The Page 5 quote included above is in reference to Military Personnel. If he is not a military member, he will file as a part-year resident of Utah, you are a non-resident. The instructions linked above are for returns when each spouse was a full-year resident of a different state.
Your situation is simpler - you will be able to file jointly, and allocate your income between the two states. Turbo Tax will generate a Part-Year Resident return for you. As you work through the Utah return, watch for a page titled 'Utah Income'. It will list your total federal income, then allow you to enter the amount sourced to Utah. (See attached screenshot.)
On your Illinois return, it will likely generate a Resident return. Respond that you would like to change that, and indicate you had more than one state of residence in 2015. This will switch you to the Part-Year/Non-Resident form, which will allow you to allocate the Illinois income in the same manner as Utah.
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