Disprove "if and only if statement"

Question

I know that this statement is false, but I am having some trouble disproving it.

If $x$ and $y$ are integers, then $x|y$ if and only if $x^2|y^2$.

Answer 1

Only if

If $x|y$, then $y=kx$ for some integer $k$, then $y^2=k^2x^2$, so $x^2|y^2$.

If

If $x^2|y^2$, consider an arbitrary prime power factor $p^k$ of $x$. $p^{2k}$ is a prime power factor of $x^2$, hence also of $y^2$. Hence, $p^k$ is also a prime power factor of $y^2$.

Since $p$ and $k$ are arbitrary, we conclude that $x|y$.