Is this true?

Recently I signed a purchase contract for a house. The seller came back saying per NJ law there is a grace period of closing dates up to 10 days, which gives them the right to postpone the closing date.

Yes, it true. "on or about" is interpreted as 10 days in either direction. Ask your attorney about your options to recoup your additional costs. At times a buyer can do a "use & occupancy" so they can move in early but not close on the home to help with moving/storage costs. As long as your mortgage extends through the cost date the seller is proposing it is reasonable to postpone under Jersey law.