This ground is packed solid from the many, many times we've run over it.
Short answer, "yes." A business owner or manager can post pretty much anything they want.
In some states, a "no guns" sign carries the force of law.
In some states, a "no guns" sign CAN carry the force of law, IF the sign meets a certain description.
In some states, the law says where you may and may not carry -- and no property owner's or manager's opinion can change that either way.
In pretty much all cases, IF you are indiscreet enough to display your weapon and are asked to leave because of company policy or the manager's private agenda, you must do so or you can be charged with trespassing.
In all cases, you have the right to refuse to patronize an establishment that treats your rights with such disdain.
Whether you acquiesce to the property owner's or manager's posted dictum (outside of your legal obligation in some states) is entirely your decision.
Some folks hold supreme the right of the property owner to have their own way in all things.
Other folks hold their own right to a privately held means of self defense to be higher still.
As you can already see in this thread, one side will usually cast around a bunch of semi-veiled insults as to the character and honor of the other side. And vice-versa. "You do it your way, and I'll do it mine. As long as you do it my way you "snuck up ol gun slob." LOL!
-Sam