steve4102 said:
...and no you do not have a "Right" to carry on private property, you may be given permission, but you cannot lose what you never had.
Think about it more closely. Private property is not all the same. There are domiciles. There are buildings, like office buildings, open to the public, where everything's generally unlocked to allow visitors, but where visitors are not common. And then there are businesses that are open to the public.
A customer doesn't have a right to be in your store; store management can kick customers out for any common sense reason, generally. But don't mistake that for a right of the business owner to take whatever arbitrary actions they see fit against customers they don't like or have ideological disagreements with. If the customer is not being disruptive in a commonly recognized way, and you pick on them anyway, you might get sued—and you'll deserve it.
Customers don't have a right, necessarily, but they should have the expectation that whatever commonly acceptable, non-disruptive behavior exists outside the doors of your business, can continue inside. If you don't like it, your best option is to politely explain to the customer why you'd rather they not do whatever it is they're doing.
There's also a necessary balancing step. If what they're doing isn't really disruptive but you don't like it anyway, and abiding by your restrictions is going to cause them grief—leaving a gun at home where it's useless, or in the glove compartment where it could get stolen, and where it's useless if there's a violent crime between their car and your store, or inside your store—your request isn't so reasonable anymore.
Pretending that property rights allow property owners to impose their will on anyone who steps onto the property, regardless of the nature of the private property, is a black and white philosophical view that has no basis in reality.