jmc

Suppose that $k=\sqrt{63-\sqrt{x}}$ is an integer. Then $0\le k\le \sqrt{63}$. 7 is the largest integer less than $\sqrt{63}$, and since $k$ is an integer, we have $0\le k\le 7$. Thus there are 8 possible integer values of $k$. For each such $k$, the corresponding value of $x$ is ${\left(63-k^2\right)}^2$. Because ${\left(63-k^2\right)}^2$ is positive and decreasing for $0\le k\le 7$, the $\boxed{8}$ values of $x$ are distinct.