Yes, we have the same (probably borrowed) phrase in English. It's real you know? Crocodolians DO emit tears when their jaws open for a strike.. to protect their eyes. It's even more sinister when the biology is put into the mix!
The great apes... I have no idea. They COULD, but my limited understanding of their behavior seems to indicate that their "crying" isn't like ours. Lacrimation as a means of communicating sorrow or distress seems to be more useful in the context of humans, who rely so heavily on facial expressions.
HOWEVER, while shedding tears may not be the primary manner you can tell their grieving, they make very clear vocalizations of what seem to be sadness, grief, distress... what you'd expect really. It seems that humans took a pretty universal adaptation, tears, and made a very short move to emphasize that as a means of greater communication.
Oh... and we don't have a ton of hair or fur, so tears tend to be VERY obvious.