First, if $Z$ is contained in the unstable locus $X^{us}=X\setminus X^{ss}$ then the ``$G$-sweep'' $G\cdot Z$ is still in $X^{us}$ since the latter is $G$-invariant. In particular, it can't be all of $Y$.

Secondly, if $Z$ is of codimension one in $Y$ then it is an irreducible component of $Y\cap X^{us}$. The latter being $G$-stable this forces $Z$ to be $G$ stable, as well. So $G\cdot Z=Z$.