Do you have to file a state return, even if you don't owe any money to the state?

Check with your state. It depends on the state and how much money you made. Some states require you to file even if you don't owe anything.

Check at the state web site for the filing requirements.

If your taxable income exceeds a certain amount (which varies from state to state), you have to file the state tax return even if the returns shows NO tax due.