Are equivalence classes closed under intersection? How do I prove whether the set of equivalence classes for some equivalence relation is closed under intersection; that is the intersection of equivalence classes is an equivalence class? 

If you're asking why... I'm trying to prove that Every two equivalence classes $[x]$ and $[y]$ are either equal or disjoint but that problem I can solve. I just need to know whether countable intersection of equivalence classes is an equivalence class.
If they are either equal or disjoint, then that means the (2-ary) intersection is either one set or the nullset. It is easily shown that this is also true for a k-ary intersection.
If I understood correctly, yes, the intersection of classes is a class. It will either be empty if you are intersecting inequivalent classes or , if not, it will agree with the first class in your intersection, which will coincide, (intersection being non-empty) with all classes being intersected..