A Private company cannot tell you what you can do in the privacy of your own home. If they tell you you cannot sing "America the Beautiful" on the front porch of your house, do you have the right?
Sure they can. Private companies can pretty much make whatever limiting rules of conduct they like. If the employee doesn't like it, the employee doesn't have to work there.
Maybe you have the right to sing all you want, but that doesn't mean the employer has to put up with it, whether you do it at home or elsewhere. After all, do you have a "right" to a job with a specific private company?
Public employers, on the other hand, have a much tougher time with this.
Nope!
But they can tell you that you're not welcome in their PRIVATE SCHOOL, if they determine you have a 'myspace' account.
Your choice!
Do you want to play on 'myspace' or go to their school?
Yes, they can (not including pre-exisiting contractual obligations, if any). In addition, my state is an at-will employment state so I can be fired for nearly anything.
A company can demand I don't smoke or do other drugs. It can fire me because it didn't like my letter to the editor. It can dump me if it discovered my secret porn career or membership in the KKK. It can demand I live within a certain geographic distance of the office place. And on and on....
A Private company cannot tell you what you can do in the privacy of your own home.