I was given a reason for being fired. Does my employer have to explain the reasoning if its not clear?

Unfortunately, unless you have an employment contract to the contrary, you may be fired at any time, for any reason--even mistaken ones. There is no wrongful termination suit arising from being fired in the absence of a contract or illegal discrmination (e.g. against a race, sex, religion, age over 40, disability status, etc.).

The company does not have to explain their reasoning or back up any claims behind their termination of you.

However, if the company publicizes accusations that you did something wrongful, and these accusations are factual, but untrue, statements, you might have a defamation suit against them.

Sorry, to clarify one thing I should have been clearer about: when I say an "employment contract," I mean an agreement in some way specifically limiting the grounds under which you can be fired. If there was such a contract, or if the confidentialility contract to which you refer contained such clauses, then you may have a cause of action. However, without an agreement which goes to the grounds for termination, then you may be terminated at will--even if it's more an incorrect claim that you violated some other contract.