05AB1E, 6 bytes

.œÅ²PO

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Explanation:

.œ      # Get all partitions of the (implicit) input
  Ų    # Vectorized check whether each inner-most value is a square
    P   # Product on each inner list, to see if all checks were truthy
     O  # Sum, to get the amount of truthy values
        # (which is output implicitly as result)

PS: Outputting the valid partitions instead of the count would be 6 bytes as well, by removing the sum O, and adding a filter ʒ earlier on: .œʒŲP Try it online or verify all test cases.

Python 2, 173 163 bytes

lambda s:len([l for l in[''.join(sum(zip(s,[','*(n>>i&1)for i in range(len(s))]+['']),())).split(',')for n in range(2**~-len(s))]if {int(x)**.5%1for x in l}=={0}])

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Edit: Saved 10 bytes due to ArnoldPalmer

Mathematica, 141 bytes

Count[FreeQ[IntegerQ/@Sqrt[FromDigits/@#],1<0]&/@(FoldPairList[TakeDrop,s,#]&/@Flatten[Permutations/@IntegerPartitions[Length[s=#]],1]),1>0]&

input (a list of digits)

[{1,6,4}]

Haskell, 88 bytes

f x=sum[0.5|y<-mapM(\c->[[c],c:" "])x,all((`elem`map(^2)[0..read x]).read).words$id=<<y]

Defines a function f that takes a string and returns a float. Very slow. Try it online!

Explanation

I'm using my Haskell tip for computing all partitions of a string with mapM and words. The snippet mapM(\c->[[c],c:" "])x replaces every character 'c' of a string x with either the one-element string "c" or the two-element string "c ", and returns the list of all possible combinations. If I take one of the results, y, concatenate it and call words on the result, it will be split at the spaces inserted by mapM. In this way I obtain all partitions of x into contiguous substrings. Then I just count those results where each partition element is a perfect square (by finding it in the list [0,1,4,9,..,x^2]). A caveat is that each partition is counted twice, with and without a trailing space, so I take the sum of 0.5s instead of 1s; this is why the result type is a float.

f x=                       -- Define f x as
 sum[0.5|                  -- the sum of 0.5 for
  y<-                      -- every y drawn from
  mapM(\c->[[c],c:" "])x,  -- this list (explained above)
                           -- (y is a list of one- and two-element strings)
  all(...)                 -- such that every element of
                 id=<<y]   -- concatenated y
          .words$          -- split at spaces satisfies this:
                           -- (the element is a string)
   (...).read              -- if we convert it to integer
    `elem`                 -- it is an element of
    map(^2)                -- the squares of
    [0..read x]            -- the numbers in this list.

Python 3, 148 139 135 134 bytes

10 bytes thanks to Arnold Palmer.

def f(a):
 s=[a[:1]]
 for i in a[1:]:s=sum([[x+[i],x[:-1]+[x[-1]*10+i]]for x in s],[])
 return sum({n**.5%1for n in x}=={0}for x in s)

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Jelly, 8 bytes

ŒṖḌƲẠ€S

A monadic link taking a list of digits and returning a non-negative integer.

Try it online! or see the test suite.

How?

ŒṖḌƲẠ€S - Link: list of digits              e.g. [4,0,0,4]
ŒṖ       - all partitions                         [[4,0,0,4],[4,0,[0,4]],[4,[0,0],4],[4,[0,0,4]],[[4,0],0,4],[[4,0],[0,4]],[[4,0,0],4],[4,0,0,4]]
  Ḍ      - convert from decimal list (vectorises) [[4,0,0,4],[4,0,   4 ],[4,    0,4],[4,      4],[   40,0,4],[   40,    4],[    400,4],     4004]
   Ʋ    - is square? (vectorises)                [[1,1,1,1],[1,1,   1 ],[1,    1,1],[1,      1],[    0,1,1],[    0,    1],[      1,1],        0]
     Ạ€  - all truthy? for €ach                   [        1,          1,          1,          1           0,            0,          1,        0]
       S - sum                                    5

Pyth, 16 bytes

lf!f!sI@sjkY2T./

Test suite.

Haskell, 135 bytes

s x=any((x==).(^2))[0..x]
c(a:b:x)=a*10+b:x
c x=x
h[x]=1>0
h x=(s.head)x
f x@(_:_:_)|y<-until h c x=f(tail y)+f(c y)
f x=sum[1|any s x]

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Probably not well golfed yet but this is a surprisingly difficult problem