You have show n implies n^2 but not the reverse.
We know that a number is either even or odd and if it's not one then its the other. We know even^2 is even, but what about the reverse?
Consider what happens when we multiply even*even and odd*odd. You can prove that even*even = even and odd*odd = odd for any odd numbers (even if they aren't the same necessarily). So this means odd*odd can't be even which means that if n = k*m and n is even then (k = ODD and m = ODD) can not occur. Similarly you have the same for even.
You can do this formally by assuming its true and then come to a contradiction (proof by contradiction is the most powerful tool in mathematics).