Your question: given that X and Z are independent, X is Gaussian (I'll use "normal"), and Y = X+Z, prove that Y is normal iff Z is normal. Right? As you observed, one direction is easy: if Z is normal, then so is Y=X+Z. So for the other direction, assume that Y is normal. We need to prove that is Z normal too.
Perhaps there's an even easier way, but it's straightforward to use characteristic functions, which completely characterise distributions. Because X and Z are independent,
$ \varphi_Y(t) = E[e^{itY}] = E[e^{it(X+Z)}] = E[e^{itX}]E[e^{itZ}]$, and so,
$ \varphi_Z(t) = E[e^{itZ}] = E[e^{itY}]/E[e^{itX}] $
This means that Z has exactly the right characteristic function for a normal variable, and hence it's normal.
More interestingly and much more generally, there is a theorem of Cramer (e.g. see here) which says that if X and Z are independent and X+Z is normally distributed, then both X and Z are!