You can brute force all possibilities easily, for example using Excel.
The square numbers below 100 are 1
, 4
, 9
, 16
, 25
, 36
, 49
, 64,
81.
Their counterparts to add up to 100 are: 99
, 96
, 91
, 84
, 75
, 64
, 51
, 36, 19.
Of these, only 19 is a prime.
So the (only) solution is: X=81 (9*9) and Y=19.