Let $G$ be a group and $m \in G$, $K$ be a subgroup and $m * m \in K$. Is it always true that $m \in K$? I'm trying to use this in another proof, but I think it might be false and I'm out of ideas to prove it. 

Not necessarily. 
Consider the rationals as a subgroup of real numbers.
$\sqrt 5 \times \sqrt 5 $ is in the subgroup but $ \sqrt 5$ is not. 
