It is possible to not owe any tax to Missouri, and even get a refund of any tax paid as a Non-Resident.
If you earned income in Missouri, prepare your Non-Resident return for Missouri to report your income earned and any taxes paid there, before preparing your Illinois Resident return.
Report all your income to Illinois, and get credit for any taxes paid to Missouri.
Here's more info:
https://ttlc.intuit.com/replies/3302052
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