I believe I should have been paying NJ withholding rather than CT withholding throughout the year but my company is telling me there is nothing they can do.
If you did not work or live in Connecticut, you do not owe any CT tax. You will have to file a Connecticut non-resident tax return in order to get a full refund of all state taxes they withheld. You will also file a New Jersey resident tax return. Since you didn't have any New Jersey tax withheld, you can expect to pay a penalty for underpayment.
I have a similar situation. I am a home office based employee working and living in NJ but the company I worked for has its headquarters in CT. The company withheld NJ taxes only. Do I need to file a non-resident CT return?
In addition to my current company in CT, I also received severance payments from another CT-based company while living in NJ. That company only withheld CT taxes (no NJ taxes). How should I file those earnings if I was not working for that company any longer, however was receiving severance payments reported as income from a CT-based company?
@novakjl -- Did you physically work in CT for the company that gave you severance pay?
CT can tax non-residents on CT-source income.
CT law is that CT-source income includes income "attributable to compensation for services performed in Connecticut or income from a business, trade, profession, or occupation carried on in Connecticut." See page 10 of this reference:
https://portal.ct.gov/-/media/DRS/Forms/2019/Income/Booklets/CT-1040NRPY-Online-Booklet_1219.pdf
Since your severance pay is attributable to work you performed in CT, it is taxable by CT.