If the square of a number is even, then the number is even. Is that true for 2?

Both $n^2$ and $n$ must be integers for this theorem to hold.

Everything you wrote is correct up until the last line. If $n^2$ is even, then $$ n^2 = 2m \quad\text{ for some } m. $$

When you wrote $n^2 = 2$, you were (falsely) assuming that $m=1$.

By the way the even/odd language is only used when you're talking about integers (whole numbers), so there is no square root of $2$.