Okay, I will take quadratic equation approach. Let’s first assume that $n$ is even $$ n^2 - n = (-1)^n + (-1)^{n^2} \\

n^2 -n = 1 + 1 \\

n^2 - n -2 = 0 \\

\text{The roots of the above equation are : 2 and -1. Since, $n$ can only be a natural number we would get $n=2$}$$

If, however, $n$ is odd, then $$ n^2 -n = -1 -1 \\

n^2 -n +2 =0 \\

\text{This equation doesn’t have any real roots}$$

Hence, $n=2$.