Its their property. They can make whatever “rule” they like. If you break their “rules” they can ask you to leave.
If you then refuse to leave, that is misdemeanor trespassing.
But carrying into a business that proclaims itself “gun free”? That’s not illegal. Their rule or policy is a mere suggestion. Treat it as such. I do.
Sometimes local and state law incorporates private designations of “gun free” zones as violations of the state or local law.
Think about this. The 2nd Amendment says “...shall not be infringed...” it does not say “...congress shall make no law...”. Technically, it says no one can infringe the RKBA. That means governments AND private entities.
Signs, signs, everywhere is signs, blocking up the scenery, messin’ my mind.