You certainly can and should use your foreign address on your tax return. The IRS and the state will send any correspondence, and mailed refunds, to the address on the tax return. So you should use an address where you will get anything that they mail to you.
You can file a New Jersey nonresident tax return showing no New Jersey income and a full refund of the New Jersey tax that was withheld. The problem you will have is that your W-2 shows New Jersey income in box 16. Because of that, the state will assume that you did have New Jersey income, but did not report it correctly. You will get a notice from the state saying that your tax return is wrong. Then you will have to try to convince them that the W-2 is wrong and that you did not work in New Jersey at all. If you can get a letter from your employer saying that you did not work in New Jersey, that would probably help. But it's a messy situation that will take time and effort to resolve, and there is no assurance of a satisfactory outcome.
(Ideally the letter from the employer would say that the New Jersey income and withholding on the W-2 are a mistake, but it doesn't sound like your employer would be willing to admit that.)
