They do have a handful of flights leaving at midnight (11:59 pm to be exact) then nothing til about 545 am. Not sure that anything will be open once those late flights depart but they should be open in the morning before the early flights.
If you are not bringing too many bags and are willing to leave the airport, you could head into Manhattan by public transportation or taxi to grab a bite to eat. There are also some 24 hour diners in Queens but you will probably have to take a taxi.
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