Do X without Y​

Ruby, 102 bytes

Array#sample doesn't do repetitions for sampling from the character set, but that's OK because the character distribution don't have to be perfectly uniform! Recursive function, returns an array of lines.

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f=->l{w,x,y,z=([*?!..?~]-%w"y Y").sample 4
l<2?[w]:[w+(s=' '*(l-2))+x,*f[l-2].map{|e|" #{e} "},y+s+z]}

Actually, 62 bytes

"!⌂"♂┘ix♂c"Yy"@-╗½≈u;r2@∙`i=`M╪k`;dXR@+`M;dXR@+`"╜J' aI"£MΣ`Mi

This is one of the longest Actually programs I've ever written.

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Explanation:

Part 1: setting up the character list

"!⌂"♂┘ix♂c"Yy"@-
"!⌂"              push the string "!⌂"
    ♂┘            CP437 ordinal of each character ([21, 127])
      ix          range(21, 127)
        ♂c        character at each ordinal (list of printable ASCII characters)
          "Yy"@-  set difference with ["Y", "y"] (printable ASCII except "Y" and "y")

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Part 2: constructing the boolean array for an X

½≈u;r2@∙`i=`M╪k`;dXR@+`M;dXR@+
½≈u;                            two copies of int(input/2)+1
    r                           range
     2@∙                        Cartesian product with itself
        `i=`M                   for each sublist: push 1 if both elements are equal, else 0
             ╪k                 split into int(input/2)+1-length chunks
                                (at this point, we have one quarter of the X)
               `;dXR@+`M        mirror each sublist (one half of the X)
                        ;dXR@+  mirror the entire list (the whole X)

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Part 3: picking random characters

`"╜J' aI"£MΣ`Mi
`"╜J' aI"£MΣ`M   for each row:
 "╜J' aI"£M        for each column:
  ╜J                 push a random value from the character list
    '                push a space
      a              invert the stack
       I             take the character if the value is 1, else take the space
           Σ       concatenate the strings
              i  flatten the list and let implicit output take care of the rest

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Mathematica, 146 bytes

a:=RandomChoice[33~CharacterRange~126~Complement~{"Y","y"}];StringRiffle[Normal@SparseArray[{{b_, b_}:>a,{b_,c_}/;c-1==#-b:>a},{#,#}," "],"
",""]&

Anonymous function. Takes a number as input, and returns a string as output.