Notice that $n$ cannot be represented as a difference of squares of integers iff $n \equiv 2 \pmod{4}$. Indeed, if $n \equiv 2 \pmod{4}$, then suppose $n = (x + y)(x - y)$. If $x, y$ are either both odd or both even, then $n \equiv 0 \pmod{4}$, otherwise $n \equiv \pm 1 \pmod{4}$. If $n = 4k$, then let $x = k + 1, y = k - 1$, if $n \equiv \pm 1 \pmod{4}$, then let $x = \frac{n+1}{2}, y = \frac{n-1}{2}$.

The strategy for Petryk is the following: he erases $100$ first, then if Ivasyk chooses $m$, then Petryk erases $100-m$. Since choice of number $50$ leads to loss, then Petryk wins, because if Ivasyk hasn't lost yet then Petryk has a number to erase that will not make him lose.