As moderators, we get occasional requests from frantic and desperate members who want their posts deleted, because they revealed too much personal information and now they are in trouble with ___________ because of stuff they posted here. So yes, it does happen. Be careful what you post on the Internet - the Internet is not as anonymous and benign as you think. Never post anything on the Internet that you wouldn't say face to face to a crowd of strangers.
You don't need to wonder. Yes, we employers monitor employees' Facebook and other social media posts.
I have dismissed 2 employees because of information they posted on fb in violation of company policy, and I printed out those pages in the event those employees attempt to file for unemployment.
My business does not involve HIPAA, but it does involve trade "secrets". Posting those on the internet is bad for one's employment prospects. So is posting pictures of yourself drinking at a party when you called off sick from work.