This is the key for me. I remember looking into this when visiting Tombstone a couple years back and seeing a bunch of no guns signs in windows.Failure to obey the request can result in your arrest for trespassing. (ARS 13-1502 / ARS 13-1503).
So, if someone inside a business asks you to remove your firearm from the premises, you must comply or risk arrest.
So the way i understand it - a no firearms sign posted in accordance with the law, next to a liquor license, in an establishment that sells liquor, legally bars a citizen from carrying open or concealed within that establishment.
Any other kind of business does not have the weight of law behind them to bar firearms from being carried in their establishment - but they do have the right to ask you to leave, at which point, if you don't, you would be guilty of trespassing.
For me, this means i would simply avoid non liquor businesses that bar firearms because i don't want to give them my money. But if i didn't see a sign, or if i needed to make a purchase and it was the only available establishment at the time, i would respectfully - as always - carry concealed with my concealed carry license, and nobody would know i was carrying.
@11Charlie appreciate the cleanup