They definitely do, Eliza. I've seen pet dogs that were very embarrassed over something that happened...either they got caught doing something they shouldn't have or some other kind of embarrassing moment occurred.

I've also seen cats that got embarrassed, usually because they slipped off something and fell unexpectedly. Cats hate that, and they really hate it if it is witnessed by anyone. We had a cat, for example, who in the enthusiasm of his youth would rush madly up and down the polished wooden stairs in our home. One day he misjudged somehow, lost his footing, and went flying down the steps and landed backwards on his ass! It was hilarious. What he did was exactly what any embarrassed cat does...he leaped up, and rushed off like a streak and wasn't seen for about half an hour. When he eventually reappeared, he walked around yawning and stretching as if to say...."Man, what a dull day. Nothing happening here. Nothing to look at here..."

That's how cats handle embarrassment. ;-D They pretend "it" never happened.