You have no case. Assuming you did not have a contract you could be fired for no reason or any reason except an unlawful reason such as based on race, religion or gender.
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You can always sue, but based on the information you provided here you would have very little chance of recovering anything.
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No, but if you think the reason they have made up a story is related to your being a minority employee, you may want to consult an attorney or the NYS DIvision of Human Rights. Otherwise, you can apply for unemployment benefits and hopefully find another restaurant to work for,.