Answer
True.
Work Step by Step
$n^2$ must be $1 \mod 4$ for odd $n$. Therefore, $n^2+1$ is $2 \mod 4$, which is not a quadratic residue. However, even $n$ are $0 \mod 4$, so adding 1 yield $1 \mod 4$ which is a quadratic residue. Therefore, even integers are the only ones that can yield perfect square $n^2+1$.
