You are definitely going to file New York, New Jersey, and Connecticut tax returns, if for no other reason than you had tax withholdings for each state, whether your income was taxable there or not. Where you live, working remotely gets quite complicated from a state tax standpoint because of each state's treatment of the "convenience to the employer" rule. Let me address the two easy states first:
New York. Since you are a New York resident, you will file a New York resident return. All of your income is taxable there no matter where you are considered to have earned it. If any portion of your income is also taxable in another state, New York will give you a credit for the tax liability you pay in that state.
Minnesota. You will not have to file a Minnesota return. The fact that the company headquarters were based out of Minnesota is irrelevant to filing taxes there. Even if you were working directly for a Minnesota office remotely, Minnesota does not employ the "convenience" principle and will not tax your income if you are not physically located in the state when you earn it.
Now for the two nonresident states you will have to file:
Connecticut. Connecticut has an interesting take on the convenience principle, which they apply specifically because of New York's policy. Connecticut's convenience principle is "reciprocal"; in other words, if the state that you are living in and teleworking from on behalf of a Connecticut business would tax a Connecticut resident working remotely for a business in that state, then Connecticut applies the convenience principle to that worker. Your situation appears to apply, because if you were physically working for the Connecticut office, Connecticut would tax you, so if you are working from home out of convenience (and COVID does not automatically make you working from home a necessity, especially because of New York's standards), then Connecticut will still tax your remote work. My recommendation is to file the income claimed to Connecticut as Nonresident Connecticut income, and any tax liability will be claimed on your New York return as a credit. (The tax liability is the amount of Connecticut tax withheld minus a Connecticut tax refund or in addition to Connecticut tax still owed).
New Jersey. New Jersey's situation is even grayer. New Jersey does not usually apply the convenience principle, but has been in a back-and-forth with New York because New York does apply it, and New Jersey loses on tax revenue with all of the Jersey residents who work in New York or telework on behalf of New York companies. If New Jersey is still maintaining that you do not pay tax to New Jersey for teleworking, then your New Jersey income will only include income earned while physically working in New Jersey. New York will provide you a credit for any tax liability you have to New Jersey.
Prepare both of the nonresident state returns first before preparing the resident return for the greatest accuracy.
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