Putting the english debate aside, I agree with firefighterchen. Not being a legal expert, but rather a believer in property owner rights, a private party has the right to put up a sign saying 'no weapons' or some such thing. However, the states have different takes as to the legal legitimacy of such signs. Some states side with the rights of the legally armed citizen, some with the property owner. Texas even went so far as to place requirements for the content of the sign, it had to site the Texas statute and be written in Spanish and English. (ok, it was over a decade ago that I took a concealed carry class in Texas, so Texas CHL holders feel free to chime in if this is out of date!).
Point being, yes the Constitution protects the right to keep and bear arms. It also protects freedom of speech. Lastly, the right of the legally armed citizen to patronize businesses that respect their rights also exists. Read the laws for your area and make a smart decision for you. Personally, I'd love for Target and Walmart to post "No Weapons" signs.... 'Sorry honey, I can't go in there, shucks..."