I am quite a beginner in group theory, so I need to get my proofs checked. Here's what I've done:
We know, that the center of a group $G$, i.e $Z(G)$ is a normal subgroup of $G$ [no proof required here]. So, the quotient group $G/Z$ can be considered. Now, consider $x^2Z*x^2Z=x^4Z=Z=(x^2Z)^2$, which readily implies that $x^2 \in Z(G)$.
Is this all? Or am I missing something?