I don't believe your circumstances qualify for leave under the FMLA. Perhaps there is a state law that will apply. Consult a local employment attorney.
Both the federal government and New Jersey have leave schemes set up for people to cope with medical emergencies. I don't see how this situation fits at all. Good luck.
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Your circumstances might be sufficiently compelling to persuade HR to grant you leave under a company bereavement policy. As a boss, I lnow that I would approve it, on an unpaid basis -- if you had no paid time left.
New Jersey Family Leave is meant for employees to take leave to care for the well-being of an immediate family member. Since that is not what you are doing here, I agree that you would have to request some sort of bereavement leave instead.