No. Let $X = E \times E$ with $E$ an elliptic curve and let $G = \mathbb Z_2 Z \oplus \mathbb Z_2$, with each factor acting on one of the $E$'s by the involution and fixing the other. The quotient is $\mathbb P^1 \times \mathbb P^1$. The effective cone of $E \times E$ is round, while the effective cone of $\mathbb P^1 \times \mathbb P^1$ is polyhedral.
I doubt there are many positive results of this sort in either direction. It's true $X$ is a Mori dream space (which implies polyhedral effective cone), then so is $Y$.