I'm not talking about them taking a dislike in you. I'm talking about you talking shit about the company or posting shit against laws or just plain offensive. If nobody can connect your profile to you/your company why should they care. If they can, it's about them as well as it's about you.
At least in my country they can't just fire you without reason. If you are a - for example - racist piece of shit on the internet and it can be tracked back to your company then hell yeah, they should at least give you a warning if not set you free.