I have a question not exactly like the OP asked.
What if a corporation said "No ford trucks on company property"
Or "No Cubs baseball hats in cars on company property"
Would there be any backlash? Would there be a winnable lawsuit there?
Guns are legal to own and keep in your cars just like the cubs hats.
There would be cause for a lawsuit IF the Cubs hat or Ford truck had a specific exception in law to grant an employee the Right to keep one in his car.
At this time there is no law of that nature so no lawsuits. However there IS a law that grants the employee the right to keep a gun on company property under specific circumstances. If the company REALLY doesn't want firearms on their property they could just tell EVERYBODY to not park on their property. Problem solved.
Also, 88GT, remember we aren't talking about a private individual telling someone "no guns". We are talking about a employer or business that has to follow specific laws on their conduct relating to employment. Private individuals who aren't subject to labor laws are completely free to ban anything they want on their property as can the owner of a business on his own personal property not related to the operation of the business.
If you want to tell blacks, mormons, naked dudes making political statements or people with guns to stay off your personal property, have at it.