I'd agree that a twitter mob getting someone fired is in most cases a bad thing and I can think of a bunch of reasons why it's a bad thing. But "infringing on the employees right to free speech" isn't one of them.

This might seem a bit pedantic, but I do think it's important. If something is a violation of someone's rights then there doesn't need to be any further discussion - it shouldn't be allowed. I think claiming that this is a free speech issue is not only wrong, but also shuts down much needed discussion about where we set the limits of our tolerance.