10+ boxes of that size is significant and you'll definitely need a cargo van or a small truck, which isn't something the major car rental companies would have.

If you're looking to do this as cheaply as possible (i.e. no professional movers), Uhaul is probably your best bet. I did a search from Trenton and it's about $120 for a one day rental, with a vehicle return just outside of JFK. The wife and kids should probably take the train and meet you at the terminal.

I'd be concerned about the optics though—I've spent my fair share of time at the airport and have never seen that many boxes at once. The boxes would literally build a wall when neatly stacked three high. I hope airport security won't give you any problems.

One thing you might look into is shipping the boxes via air freight with the airline (rather than something like FedEx). It'll probably go on the same plane and they have a dedicated freight drop-off area. The extra complexity with customs might not make it worth it though.