Just because a store/mall is open to the public does not make it "public property." It is still privately owned.
Yes, they can enforce trespassing on any person for any reason, and they actually don't even have to tell you why. In this case, they do have the authority to tell people that they cannot bring in firearms.
So the best thing to do is leave and write to the stores that you were going to spend money in to tell them why you did not.
The signs themselves are legally meaningless here, though. You do not have to look for them, nor should you.