To find all integers n such that the given expression is a perfect square

Everything you did was correct, but you missed the factorization $255=1\cdot 255$. This gives $4k=1+255=256$ and so $k=64$, giving $$n^4-n^2=k^2-64=64^2-64,$$ which gives that $n=\pm 8$ -- exactly what you missed.