This question seems to mix up a couple of different rules. First, as an American citizen you can go where you want. But second, the consequence could be that you lose your spot in the facility. My guess is that your care is covered by Medicaid. When a Medicaid recipient leaves a nursing home to visit family it is called "therapeutic leave" and states differ in how long their Medicaid programs will pay to hold a bed for a resident on therapeutic leave; some won't pay anything while others will pay up to 30 days.
If you are absent longer than what New Jersey Medicaid allows (which is 24 days, according to one source), the facility will not be paid. If it is not paid, it may have the right to evict you for nonpayment. The safest bet would be for your family to pay the facility during the additional days you are absent. You may be able to negotiate payment at the lower Medicaid rate rather than at the higher private cost per day.
For an article on leaving a nursing home while on Medicare or Medicaid (including a link to state bed-hold policies), click here.
For more on the rights of nursing home residents, click here.
For an article on how to fight a nursing home discharge, click here.