Your property, you do what you want. Doesn't mean I agree with it but as long as they aren't breaking a law they can do whatever they want.
Well as a former employee of publix I'm pretty sure that they lease that building from a seprate party, but if they do own it wouldn't the company as a whole have to make the decision not just the store manager since he/she don't own that store. I carry into my local publix without any problems. Sometimes even open carry.