...I have no problem with property owners of mostly any size property posting a "No Guns" request/statement/demand. I feel the public may be due a bit of diligence before you post a sign establishing your venue, be it a corner bar or Giants Stadium, as "Gun-Free".
I work as a Systems Engineer in some industries that have extensive government regulation. Before certain products are acceptable, the manufacturer/developer has to exercise a bit of auditable diligence to validate conformance to standards and verify critical functionality. Safety organizations like UL and CSA have standards and performance requirements for various products. I'm thinking that the actual Gun-Free signs should mean something to those just and honest folks that patronize a store, business or venue. They should at least mean more than the owner had $10 to buy a sign.
I feel any owner can make the rules for their property. "No shirt, no shoes, no service", I'm good with that. "No guns allowed", fine with that, too. You have to tell people what you need. On the other hand the sign "Gun Free Zone" looks to me to more of an indication of the owner/management taking a role in assuring a certain type of security and more of characterization than a request to the patron, employee or visitor.
Just another example: yellow and black sign on the road the reads CAUTION 35 MPH means we the government request you not exceed this speed; a black and white sign that reads SPEED LIMIT 35 MPH means go 40 here and we'll fine your ass if we catch you.