Here at Page Lizard, we haven't experienced negative Facebook comments (praise the Lizard!) but we have seen it happening all over the web.
As the brand has matured, so has its online tactics. Today Walmart takes the time to respond to all customer feedback, be it negative or positive. Why not have a chance to defend yourself? Why not offer an unsatisfied customer a solution?
If you fail to respond to negative Facebook comments on your fan page then it will appear to other customers that you are just trying to ignore the issue. Don't be that guy! Being unresponsive does nothing more than entice the user to come back bigger than ever. Give them the time of day, even if you think their accusations are false.
My top tip for dealing with Facebook negativity is to take the battle elsewhere. Don't have it out with a user across your Facebook page. Offer up your company email address so you can resolve the issue away from the public eye.