Brian Conrad says: "First reduce to the case when Y is also finitely presented over Z. Then you can use limit arguments, so it becomes an easy application of basic facts about strictly henselian local rings and finite algebras over them. It will be instructive for you to think about it some more for yourself in view of these hints.
I should have also added that you must be assuming $X\to Y$ is surjective, or else it clearly isn't true. That condition enters into the argument when doing analysis of the situation with strictly henselian local rings."