To actually produce tears the Lacrimal gland is used, as far as I know all mammals (specially common mammals) have a Lacrimal apparatus system and therefore can cry.
Remember tho, just like a human. If an animal has tears in its eyes it doesn't necessarily mean it is sad, it could just have something irritating its eye.
Determining if an animal cries because it is sad is much harder. What I do know is that pain can set of tears. Sadness is a pain right?
Hope that helps