jmc

$46656=2^6\cdot 3^6$. So, $x$ is a factor of $46656$ if and only if there exist integers $a$ and $b$ such that $0\le a\le 6$, $0\le b\le 6$, and $x=2^a{3}^b$. Such an $x$ is a perfect square if and only if $a$ and $b$ are both even, which is true if and only if $a$ and $b$ are each 0, 2, 4, or 6. There are $4$ possible values of $a$, and $4$ possible values of $b$, so there are $4\cdot 4=\boxed{16}$ possible values of $x$.