New York: if you're a non-resident of NY, and assuming you're working remotely for your own convenience, rather than because you were directed to do so by your employer, ALL your income from that work is taxable by New York. This is called the "convenience of the employer" doctrine, which is employed by a small number of states.
New Jersey: ALL your income during the time you were domiciled in NJ is (also) taxable by NJ.
California: ALL your income during the time you were domiciled in CA is (also) taxable by CA.
Therefore at year's end you'll be filing:
1. a non-resident tax return for NY, reporting all your remote earnings; and
2. a part-year resident tax return for NJ, reporting all the income you had as an NJ resident; and
3. a part-year resident return for CA, reporting all the income you had as a CA resident.
You will be able to claim an "other state credit" on both your NJ and on your CA return for the taxes you paid to NY on the income also taxed by the particular resident state, so you won't be double-taxed. Thus, because of the credits you can take, most (if not all) of your net state taxes due will be due to NY.
So you definitely want your employer to withhold NY taxes. Your credits for NY may or may not cover your taxes due to the other states, depending on the numbers. But your entire year's remote-work income will be taxable by NY.
Post back if you were required to work remotely by your NY employer, because that would totally change the scenario.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
