J-uby, 47 bytes

~:^%(:**&10|:+|:select+~(S**:[])%~:*.D|:& &:[])

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Vyxal 3, 8 bytes

kṬʎ²nc}i

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kṬʎ²nc}i­⁡​‎‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁣‏⁠‎⁡⁠⁢⁣‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏‏​⁡⁠⁡‌­
       i  # ‎⁡index implicit input from list
kṬ        # ‎⁢all integers
  ʎ   }   # ‎⁣filtered by
   ²nc    # ‎⁤does the number squared contain it?
💎

Created with the help of Luminespire.

<script type="vyxal3">
kṬʎ²nc}i
</script>
<script>
    args=[["0"],["1"],["2"],["3"],["4"],["5"]]
</script>
<script src="https://themoonisacheese.github.io/snippeterpreter/snippet.js" type="module"/>

Tcl, 70 bytes

proc S n {while {[incr j [regexp [incr i] [expr $i**2]]]-$n} {}
set i}

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AWK, 33 bytes

{for(;j<$1;)(x=i^2)~i++&&j++}$0=x

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Japt, 12 11 8 bytes

1-indexed

ȲsèX}iU

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ȲsèX}iU     :Implicit input of integer U
È            :Function taking an integer X as argument
 ²           :  Square
  s          :  Convert to string
   èX        :  Count the occurrences of X
     }       :End function
      iU     :Get the Uth 0-indexed non-negative integer that returns truthy (not 0) when passed through that function

☾, 14 chars (43 bytes)

󷺹!3⭥󰈲󷺹󷹜∈󷹜⟞²ᐸ→􍪴

Note: ☾ uses a custom font, but you can run this code in the Web UI

Go-go gadget galactic alg

cQuents, 8 bytes

#SN)iSNN

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Explanation

         implicit mode : sequence 1, given input n output the nth term
#        N is in the sequence if
 S )                             string (   )
  N                                       N
    i                                         in 
     S                                           string (       )
      NN                                                  N * N

Tidy, 24 bytes

{x:str(x)in'~.x^2}from N

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Returns a lazy list which, when called like a function, returns the nth element in the series.

Explanation

{x:str(x)in'~.x^2}from N
{x:              }from N       select all natural numbers `x` such that
   str(x)                      the string representation of `x`
         in                    is contained in
           '~.x^2              "~" + str(x^2)

Lua, 137 123 79 bytes

-thanks @Jo King for 44 bytes

n=io.read()i=0j=0while(i-n<0)do j=j+1i=i+(n.find(j*j,j)and 1or 0)end
print(j*j)

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Clojure, 81 bytes

(fn[n](nth(filter #(clojure.string/includes?(str(* % %))(str %))(range))n))

Try it online! (Unfortunately, TIO doesn't seem to support Clojure's standard string library)

If Clojure had shorter importing syntax, or had a includes? method in the core library, this could actually be somewhat competitive. clojure.string/includes? alone is longer than some answers here though :/

(defn nth-sq-subs [n]
  (-> ; Filter from an infinite range of numbers the ones where the square of
      ;  the number contains the number itself
    (filter #(clojure.string/includes? (str (* % %)) (str %))
            (range))

    ; Then grab the "nth" result. Inc(rementing) n so 0 is skipped, since apparently
    ;  that isn't in the sequence
    (nth (inc n))))

Since the TIO link is broken, here's a test run. The number on the left is the index (n), and the result (N) is on the right:

(mapv #(vector % (nth-sq-subs %)) (range 100))
=>
[[0 1]
 [1 5]
 [2 6]
 [3 10]
 [4 25]
 [5 50]
 [6 60]
 [7 76]
 [8 100]
 [9 250]
 [10 376]
 [11 500]
 [12 600]
 [13 625]
 [14 760]
 [15 1000]
 [16 2500]
 [17 3760]
 [18 3792]
 [19 5000]
 [20 6000]
 [21 6250]
 [22 7600]
 [23 9376]
 [24 10000]
 [25 14651]
 [26 25000]
 [27 37600]
 [28 50000]
 [29 60000]
 [30 62500]
 [31 76000]
 [32 90625]
 [33 93760]
 [34 100000]
 [35 109376]
 [36 250000]
 [37 376000]
 [38 495475]
 [39 500000]
 [40 505025]
 [41 600000]
 [42 625000]
 [43 760000]
 [44 890625]
 [45 906250]
 [46 937600]
 [47 971582]
 [48 1000000]
 [49 1093760]
 [50 1713526]
 [51 2500000]
 [52 2890625]
 [53 3760000]
 [54 4115964]
 [55 5000000]
 [56 5050250]
 [57 5133355]
 [58 6000000]
 [59 6250000]
 [60 6933808]
 [61 7109376]
 [62 7600000]
 [63 8906250]
 [64 9062500]
 [65 9376000]
 [66 10000000]
 [67 10050125]
 [68 10937600]
 [69 12890625]
 [70 25000000]
 [71 28906250]
 [72 37600000]
 [73 48588526]
 [74 50000000]
 [75 50050025]
 [76 60000000]
 [77 62500000]
 [78 66952741]
 [79 71093760]
 [80 76000000]
 [81 87109376]
 [82 88027284]
 [83 88819024]
 [84 89062500]
 [85 90625000]
 [86 93760000]
 [87 100000000]
 [88 105124922]
 [89 109376000]
 [90 128906250]
 [91 146509717]
 [92 177656344]
 [93 200500625]
 [94 212890625]
 [95 250000000]
 [96 250050005]
 [97 289062500]
 [98 370156212]
 [99 376000000]]

This should be able to support any value of n; providing you're willing to wait for it to finish (finding the 50th to 100th integers in the sequence took like 15 minutes). Clojure supports arbitrarily large integer arithmetic, so once numbers start getting huge, it starts using BigInts.

Perl 5 -p, 33 bytes

map{1while++$\**2!~/${\}/}1..$_}{

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APL (Dyalog Extended), 31 30 bytes

1∘{>⍵:⍺-1⋄(⍺+1)∇⍵-∨/(⍕⍺)⍷⍕⍺×⍺}

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0-indexed.

Python 2, 47 43 bytes

^{-4 bytes thanks to Dennis (adding 1 to recursive call instead of returning n-1)}

f=lambda c,n=1:c and-~f(c-(`n`in`n*n`),n+1)

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Explantion/Ungolfed

Recursive function taking two arguments \$c,n\$; \$n\$ is counts up \$1,2,3\dots\$ and everytime \$n \texttt{ in } n^2\$ it decrements \$c\$. The recursion ends as soon as \$c = 0\$:

# Enumerating elements of A018834 in reverse starting with 1
def f(counter, number=1):
    # Stop counting
    if counter == 0:
        return 0
    # Number is in A018834 -> count 1, decrement counter & continue
    elif `number` in `number ** 2`:
        return f(counter-1, number+1) + 1
    # Number is not in A018834 -> count 1, continue
    else:
        return f(counter, number+1) + 1

Haskell, 60 bytes

([n^2|n<-[1..],elem(show n)$words=<<mapM(:" ")(show$n^2)]!!)

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      n<-[1..]              -- loop n through all numbers starting with 1
 [n^2|        ,    ]        -- collect the n^2 in a list where
     elem(show n)           -- the string representation of 'n' is in the list
       words ... (show$n^2) -- which is constructed as follows:

            show$n^2        -- turn n^2 into a string, i.e. a list of characters
          (:" ")            -- a point free functions that appends a space
                            -- to a character, e.g.  (:" ") '1' -> "1 "
        mapM                -- replace each char 'c' in the string (n^2) with
                            -- each char from (:" ") c and make a list of all
                            -- combinations thereof.
                            -- e.g. mapM (:" ") "123" -> ["123","12 ","1 3","1  "," 23"," 2 ","  3","   "]
      words=<<              -- split each element into words and flatten to a single list
                            -- example above -> ["123","12","1","3","1","23","2","3"]

(                      !!)  -- pick the element at the given index

Java 8, 66 65 63 bytes

n->{int r=0;for(;!(++r*r+"").contains(r+"")||--n>0;);return r;}

-1 byte thanks to @Shaggy.
-2 bytes thanks to @Arnauld.

1-indexed.

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Explanation:

n->{                // Method with integer as both parameter and return-type
  int r=0;          //  Result-integer, starting at 0
  for(;             //  Loop as long as:
       !(++r*r+"")  //    (increase result `r` by 1 first with `++r`)
                    //    If the square of the result `r` (as String) 
        .contains(  //    does not contain
          r+"")||   //    the result `r` itself (as String):
       --n>0;);     //     (decrease input `n` by 1 first with `--n`)
                    //     And continue looping if input `n` is not 0 yet
  return r;}        //  Return the result `r`

Ruby, 45 bytes

->n,i=1{/#{i+=1}/=~"#{i*i}"&&n-=1while n>0;i}

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Edit (response to comments): Python 2, 76 bytes

Wanted to try for a non recursive method. (New to golfing, any tips would be great!)

def f(c,n=0):
    while 1:
        if`n`in`n*n`:
            if c<2:return n
            c-=1
        n+=1

Thanks both BMO and Vedant Kandoi!

Jelly, 6 bytes

1ẇ²$#Ṫ

1-indexed.

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How?

Finds the first n of the sequence as a list and then yields the tail, N.

1ẇ²$#Ṫ - Link: integer, n (>0)
1      - initialise x to 1
    #  - collect the first n matches, incrementing x, where:
   $   -   last two links as a monad:
  ²    -     square x
 ẇ     -     is (x) a substring of (x²)?
       -     (implicitly gets digits for both left & right arguments when integers)
     Ṫ - tail

If 0 were considered a Natural number we could use the 1-indexed full-program ẇ²$#Ṫ for 5.

Perl 6, 33 31 bytes

-2 bytes thanks to nwellnhof

{(grep {$^a²~~/$a/},1..*)[$_]}

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Explanation:

{                            }  # Anonymous code block that returns
 (                      )[$_]   # The nth index of
  grep {          },1..*        # Filtering from the natural numbers
        $^a²                    # If the square of the number
            ~~/$a/              # Contains the number

Charcoal, 25 bytes

Ｎθ≔¹ηＷθ«≦⊕η¿№Ｉ×ηηＩη≦⊖θ»Ｉη

Try it online! Link is to verbose version of code. 0-indexed. Explanation:

Ｎθ

Input n.

≔¹η

Start N at 1. (Or, this could start counting at 0 which would make the input 1-indexed.)

Ｗθ«

Repeat until we have found n numbers in the sequence.

≦⊕η

Increment N.

¿№Ｉ×ηηＩη

If N*N contains N, then...

≦⊖θ»

... decrement n.

Ｉη

Print N.

My attempts at golfing this further were stymied by Charcoal a) not having an if..then except at the end of a block (which costs 2 bytes) b) not having a Contains operator (converting the output of Find or Count into a boolean that I could subtract from n again costs 2 bytes).

Common Lisp, 95 bytes

(lambda(n)(do((i 0))((= n 0)i)(if(search(format()"~d"(incf i))(format()"~d"(* i i)))(decf n))))

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Clean, 83 bytes

import StdEnv,Text

(!!)[i\\i<-[1..]|indexOf(""<+i)(""<+i^2)>=0]

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(!!)                               // index into the list of
 [ i                               // i for every
  \\ i <- [1..]                    // i from 1 upwards
  | indexOf (""<+i) (""<+i^2) >= 0 // where i is in the square of i
 ]

MathGolf, 8 bytes (works for any input in theory, but only for n<10 in practice)

úrgɲï╧§

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Alternative (works for n<49 in practice and theory)

►rgɲï╧§

The only difference is that instead of creating a list with 10^(input) values, I create a list with 10^6 items. This takes a while to run, so you could swap the first byte to any other 1-byte literal to test it out.

Explanation

ú          pop(a), push(10**a)
 r         range(0, n)
  g        filter array by...
   É       start block of length 3
    ²      pop a : push(a*a)
     ï     index of current loop
      ╧    pop a, b, a.contains(b)
           Block ends here
       §   get from array

The reason why this solution doesn't handle large input is that I noticed that the sequence grows less than exponentially, but more than any polynomial. That's why I used the 10**n operator (I wanted to use 2**n but it failed for input 1). That means that I create an extremely large array even for small inputs, just to filter out the vast majority of it, and then take one of the first elements. It's extremely wasteful, but I couldn't find another way to do it without increasing the byte count.

JavaScript (ES6), 43 bytes

f=(n,k=0)=>n?f(n-!!(++k*k+'').match(k),k):k

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Non-recursive version, 47 bytes

n=>eval("for(k=0;n-=!!(++k*k+'').match(k););k")

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05AB1E, 6 bytes

1-indexed

µNNnNå

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Explanation

µ         # loop over increasing N until counter equals input
 N        # push N (for the output)
  Nn      # push N^2
    N     # push N
     å    # push N in N^2
          # if true, increase counter