I've NEVER heard of anyone having a problem leaving their car at JFK. The only thing I would suggest is since you're going to be gone for such a long time, is taking public transportation. I don't know NJ that well, are you in an area where NY Penn Station is accessible? Then just take LIRR to Jamaica/Air Train.
P.S.: How dare you post a travel related thread on Flyertalk in OMNI!!!
