I'm getting taxed $1,058 in Illinois but I have no taxable Illinois income. My only income was from the Military.

Yes, you are correct active duty military pay is not taxed in Illinois. See Pub102 Illinois Filing Requirements for Military Personnel for detailed information.

Please ensure that you have entered Illinois in the State that you lived in in the "My Info" section even when you were stationed somewhere else:

1. Login to your TurboTax Account 
2. Click "Work on other tax breaks"
3. Click "My Info" from the left side of your screen
4. Click “Edit” next to your name
5. Please scroll down to “2. Tell us the state(s) you lived in” 

Also, make sure that you checked the active-duty military box for the W-2 on the "Do any of these uncommon situations apply to this W-2?" screen right after you enter your W-2.

Furthermore, when you get to your state return double-check that your military pay is entered correctly on the "Military Pay Exclusion" screen for Illinois.