If an employee does not divulge in any way their connection to an employer, then I don’t see that the employer can take action short of an employee admitting to the commission of a felonious act. If the employer can monitor and regulate the behavior of their employees away from the workplace, then we are into new territory.
Can your employer fire you for going to church? What about for advocating the arming of teacher and other staff at public schools? What if you run a Tea Party web site?
One common feature in all the cases mentioned is that the person obviously connected themselves with the business since they cost the company customers.