As for your argument that, "We may as well as make it impossible for any business to fire people for any reason, then", I don't see how that follows from what I am proposing. Firing people for reasons that relate to their work makes sense, firing someone for making an OK sign to someone on the highway does not. Someone says "Studies show that violent protests decrease support for the protestors' cause" at work, feel free to walk them out the door for their hate speech. If they say it on Reddit, maybe that's not the hill to die on.
I don't think whether a job is a right or not is the question we should be asking. People have obligations, both to their families, and to society. Someone can't get a job because of something they said on the internet and now they contribute nothing to society even though they are perfectly capable of being productive. That doesn't make sense. I pay more taxes now because they aren't working?
You're going to tell some kid, "Your mom doesn't get child support anymore because your dad said something mean on the web"? How does that even make sense?