Consider a = 5, then a^2 = 25, but 25 is not even. This means we have reached a contradiction. Hence we can conclude no perfect square is congruent to 2 mod 4.
The part "this means a^2 for any a in Z must be even" was underlined and no was written next to it. When I said "we have reached a contradiction" was also underlined and written no. So I'm skeptical to now think if what I found wasn't a contradiction, but the proof seems correct to me, so it is possible that the grader thought I was doing a proof by example.