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Take it from me... I did this Halloween night (which was also the end of daylight savings which added an additional hour to my wait)

The above is correct. All the JFK terminals close except T4. The arrivals hall is the place to be. There is an Au Bon Pain and a Hudsen News open all night. There were about 100 or more other people spending the night as well. Seating is very limited, and I doubt you can get any real sleep so I suggest trying to stay awake to sleep well on the flight in the morning. The AirTrain also operates all night and can get quite comfy if it stays empty. The trek to Jamaica and back takes about 25 minutes so doing that many times can pass some time.

As for T7, the doors open at 4am sharp and the UA machines are turned on at 430am. Security open sometime before 5am.

It's a long wait, but doable. I regretted not taking my laptop. That would have made the wait 100X better, so if you have one, don't make the same mistake.