I don't think it's a free speech issue, because people aren't free from social consequences for what they say. And I think people should anticipate that anything they say on a public website under their real name could be seen by their boss or other people whom they'd rather not have read it.
That said, I'm not entirely comfortable with the internet vigilantism aspect, mainly because I think people can be prone to jumping on a bandwagon without thinking things through or verifying the story. There have been examples of people faking inflammatory stuff about exes or other people they dislike and using the internet to try to get revenge, so it can be hard to know, sometimes.
I also think that people need to consider what they're trying to accomplish. Though I don't have much sympathy for people who say racist stuff on social media, I do question whether going after these people individually does much good, unless the individuals are in a position of power or their remarks are related to their job somehow.
It's less that I think that these people being fired is right or wrong, but more that the simple logic of "everything you post on the internet is public forever, therefore I have the right to send anything and everything straight to your boss and am completely morally clean of any consequences, see YOU brought this on YOURSELF for DARING to say anything that would potentially get you fired ever" is very flawed and that there must be a better way to justify it, if you do want to justify it.
This is even more true when you say the internet is forever but consider that people's opinions are not.
I agree with this. I think people should be cautious about what they say/do online, but some of that is deciding what risks you're willing to take and what type of speech you're willing to stand by.
The thing is, there are a lot of things you could say that could potentially cause trouble with someone. Even stuff that's harmless. I live in a state where there's no protection against anti-LGBT discrimination, so one of my concerns is that if a potential employer googles me, they might be able to find some stuff related to my participation in LGBT-related groups. But I know I have nothing to be ashamed of about that, and I don't want to have to go out of my way to hide that part of my life, so that's a risk I'm willing to take.
There are also things, like my erotica and interest in kink, that I ideally don't want my family or workplace to know about. But I talk about that stuff under pseudonyms, and I figure if I was outed somehow, I would be willing to say, "Hey, this is another part of my life that I've intentionally kept separate, but I'm not ashamed of it in any way."
The key, I think, is being willing to stand by what you say/do if necessary, and to make smart decisions about how open and public you are.
So I agree with you that it's not a black and white thing. Some people say things that are truly vile, but people can get also get in trouble because their
boss is racist, homophobic, or otherwise prejudiced. I think making good decisions about what image you project online is more complicated than telling people to never say anything they wouldn't want their boss/grandmother to see.