Don't check the box. The amount that is reported on that screen is the net state tax that is being calculated for IL credit. (In general, this is the MO state withholdings minus MO refund or added on additional MO state tax owed). The box you see is a special provision for local taxes withheld by a reciprocal state of Illinois (MO and IL are not reciprocal). IL and KY are reciprocal. An Illinois resident who works in Kentucky does not have KY state tax withheld, but may have KY local tax withheld. Checking that box allows the Illinois resident to claim the KY local tax (which is not covered in the reciprocal agreement) for a credit on their Illinois state return.
In your case, the box does not apply because your Missouri tax credit is probably zeroing out Illinois tax. Even if this is not the case, however, the 1099G refund amount is not included in the Tax Paid to Missouri figure because that is your state tax refund from last year. Bottom line is that if your Illinois tax is zero, you will not check the box and you will not need to enter anything on the screen related either to your local taxes withheld or your MO tax refund.
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