If you leave the U.S. you will need a valid J-1 visa stamp to reenter this country (unless you are Canadian). Do not assume that with an approved extension of stay you can reenter this country from another continent without a valid J-1 visa stamp in your passport (unless you are Canadian). If yours has expired, you will have to apply for a new one at a United States consulate abroad. Your dependents will need J-2 visas, and you should be prepared to show proof of marriage to your spouse and parenthood of each child.
CAUTION: Extension of your permission to stay is your responsibility. If you are employed and overlook the date, you will be working illegally. Since such mistakes can have serious consequences, you should make certain that you apply well in advance (three months is recommended) if you need to extend your stay.