Yes, I recognize there are some narrow exceptions such as libel, and I'm perfectly fine being held accountable for libel. If I post something which actually libels someone, i.e. something which a court will hold me accountable for having posted, then that's fair, and I'll take full responsibility.
What seems to be in question here, however, is whether people may post truthful comments on twitter, which under U.S. law are by definition not libelous.
It's more than just libel though. While truthful, your employer might not like being associated with the opinions that you're sharing. That's why employers have social media policies - to protect themselves by educating their employees so that they understand that the employee can be held accountable for the opinions shared via social media if that employee is identifying themselves online as being associated with that company. It is extremely common, and I would suspect your University does have an opinion on the matter - especially if it's a State funded school. Then especially, you're a State employee and are representing that larger group too.
What seems to be in question here, however, is whether people may post truthful comments on twitter, which under U.S. law are by definition not libelous.