Jelly, 34 bytes

Ḃr1Ḥṁa5B¬¤7_BZ«ḷþm2Ɗṙ1Ṛ;UƊṁṙHĊ×ḂƲḂ

A monadic Link that accepts the side length (\$\ge 5\$) and yields a list of lists of ones (white / clue area) and zeros (black / background).

Try it online! (The footer formats each as a grid, which spaces them to be more square, replace G with Y if you prefer no spaces.)

(Admittedly, they're not very intricate, so they are not that pretty, but they adhere to the specification.)

How?

Ḃr1Ḥṁa5B¬¤7_BZ«ḷþm2Ɗṙ1Ṛ;UƊṁṙHĊ×ḂƲḂ - Link: integer, Side
Ḃ                                  - {Side} mod 2
 r1                                - inclusive range to 1 -> odd:[1] even:[0,1]
   Ḥ                               - double -> [2] or [0,2]
    ṁ                              - mould like {[1..Side]}
                                      -> Reps = [2,2,2,...] or [0,2,0,...]
         ¤                         - nilad followed by link(s) as a nilad:
      5                            -   five
       B                           -   to binary
        ¬                          -   logical NOT -> [0,1,0]
     a                             - {Reps} logical AND {[0,1,0]}
                                      (replace first three of Reps with [0,1,0])
          7_                       - seven minus {that}
                                      -> [7,6,7,5,5,5,5,5,...,5,5] if Side is odd
                                         [7,6,7,5,7,5,7,5,...,7,5] if Side is even
                                      ...of length Side.
            BZ                     - convert to binary and transpose
                                      -> RowsAB&C
                   Ɗ               - last three links as a monad - f(Side):
               ḷþ                  -   {[1..Side]} left-arg {[1..Side} table
                                        -> M = [[1..Side]] * Side
                  2                -   two
                 m                 -   {2} slice {M}
                                        -> H = [[1..Side]] * ((Side + 1) // 2)
              «                    - {RowsAB&C} minimum {H} (vectorises)
                                      -> H with RowsAB&C as the first three rows
                    ṙ1             - rotate left one (moving Row A to the bottom)
                      Ṛ;UƊ         - reversed concatenated with each-reversed
                          ṁ        - mould like {[1..Side]} (remove last row if odd)
                                      -> PuzzleRows
                                Ʋ  - last four links as a monad - f(Side):
                            H      -   halve {Side}
                             Ċ     -   ceiling of {that}
                               Ḃ   -   {Side} mod 2
                              ×    -   {Ceil(Side/2)} multiply {Side mod 2}
                           ṙ       - rotate {PuzzleRows} left by {that}
                                 Ḃ - mod 2

Charcoal, 58 bytes

ＮθＭ﹪θ²↑ＵＯ⊘⊕θO#¶OOＭ﹪θ²↘×#⊖⊘θ⟲ＯＯ²⁴⁶﹪θ²Ｊ⊘⊖θ⊖⊘θ≡﹪θ⁴¦⁰O#¶#³#↘##

Try it online! Link is to verbose version of code. Uses Os instead of _s as I think it looks nicer. Explanation:

Ｎθ

Input N.

Ｍ﹪θ²↑ＵＯ⊘⊕θO#¶OO

Draw a cross-hatch to fill the top left ¼ of the crossword. For odd sizes, the cross-hatch starts at the corner, but for even sizes it's vertically offset. (The code actually performs the reverse operation, drawing an offset cross-hatch by default but undoing the offset for odd sizes.)

Ｍ﹪θ²↘×#⊖⊘θ

For even sizes, erase the first line, while for odd sizes, erase all but the first character on the second line.

⟲ＯＯ²⁴⁶﹪θ²

Rotate to mostly complete the other ¾ of the crossword.

Ｊ⊘⊖θ⊖⊘θ≡﹪θ⁴¦⁰O#¶#³#↘##

Fix up some edge cases. (Well, they're in the centre, not on the edge, but you know what I mean.)