If $n$ is odd, then both $a$ and $b$ are odd and therefore $n = a^2 + b^2$ is even, contradiction. Therefore, $n$ is even and $a = 2$. This also shows that $b$ is even. Furthermore, $b \mid (n - b^2) = a^2 = 4$. Thus $b \in \{2, 4\}$, which results in $n = 8$ and $n = 20$, respectively.

If $n$ is odd, then both $a$ and $b$ are odd and therefore $n = a^2 + b^2$ is even, contradiction. Therefore, $n$ is even and $a = 2$. This also shows that $b$ is even. Furthermore, $b \mid (n - b^2) = a^2 = 4$. Thus $b \in \{2, 4\}$, which results in $n = 8$ and $n = 20$, respectively. $\square$