Their insurance company, on the other hand, can make them do anything it wants.
Insurance carriers do not set specific criteria for each state when writing business. For example, they won't take into consideration that one state bans firearms on premises where alcohol is sold for consumption on premise, versus a state that does not ban such. All that typically matters to an insurance company is that the question is asked "Do you have firearms on premises?" And that question is not directly pointed to actions of the insureds customers, its about the insured and their employees.
I attended an OSHA conference some years back and was able to ask the speaker about businesses that post signs like the one shown at this AL. His response was that its only a 'feel good measure' that makes the company more comfortable. He likened it to the signs at a carnival stating 'you must be xxxx tall to ride'. That doesn't mean that someone who is shorter than the height recommended will be denied coverage in the event of a loss. And it doesn't necessarily mean someone who is armed will be denied coverage in the event of a loss on a premise where signs indicate possession of a firearm is prohibited.