I plan on working for a US company from abroad for three months. Do I need a US address at this time to remain a tax resident of my state?

Interestingly, say you sold your home and moved overseas FOREVER !

Your tax residence location would still be NJ.

Here's a clear explanation re US'ers who live overseas for a long time:

https://vhd.overseasvotefoundation.org/index.php?/ovf/Knowledgebase/Article/View/1030/1/which-state-do-i-vote-in-if-i-no-longer-maintain-us-residence

If you holiday overseas for three months,

and indeed "get rid of" your rental or owned home in NJ when you leave,

there are absolutely no US tax implications, whatsoever. You do not even "tell them" about it; it is fortunately completely irrelevant.

The one and only change would be, on your next tax return, you'd have "changed address" - that's it. Nothing else.

I'm familiar with being a part-year state resident in a a state that also collected income tax. Looking over New Jersey it appears to be the same. That is they look at your intent for the move as to whether it is temporary or permanent. From your situation, the intent definitely appears to be temporary and you'd be a full-year resident of NJ as a result. If you could document your intention to stay long term in another state or country (driver's license, residency permit, bank accounts, etc.), you could make the case that the move was indeed intended to be permanent, and then would not be subject to NJ state income tax for the period you did not live there.

For New Jersey Income Tax purposes, your residency status depends on where you were domiciled and where you maintained a permanent home during the tax year. Generally, when you change your domicile during the year, you are a resident of New Jersey for part of the year (part-year resident) and a nonresident of New Jersey for part of the year (part-year nonresident). Your move is generally considered a change of residency status (resident to nonresident or vice versa) if, at the time you moved, you intended to permanently leave one home and establish a new permanent home somewhere else.

In regards to US income tax, you'd still have to file that the same as you previously did as the length of stay outside the US wouldn't allow for you to be eligible the Foreign Earned Income Exclusion.