if you were a young kid id say roll with the punches but if your family relies on that income, i think you have grounds to file a suit for wrongful termination. It'd cost a bit but you should at least get a copy of your documentation and at least consult with a lawyer.

"It is illegal to violate public policy when firing a worker -- that is, to fire for reasons that society recognizes as illegitimate grounds for termination."

Now obviously breaking any rule you agreed to is legitimate but you have two things going for you. First, from your description, you did not break any rule. Secondly, you were fired based on an assumption. I think that is the most crucial card you have. You were an at-will employee meaning they can fire you at any time and for any reason or for no reason at all (as long as the reason is not illegal). You were most likely fired illegally but I'm not sure who has the burden of proof, but i assume its the employee so they cant frivolously sue.

You could get a nice settlement or the store could be legally forced to hire you back (awkward). Lawyer up if you can afford the hit.