You do not indicate whether your son had any taxable income while living in Illinois. If he had income above the state's filing threshold, he should file an Illinois part-year resident or nonresident tax return, depending on where his domicile was while he was in school.
If he was living and working in Wisconsin during thew latter part of the year, he should file a Wisconsin part-year resident return, assuming that his wages from the new job place him above the filing threshold.
For state-specific information, your son should refer to the state taxing authority websites:
Illinois Department of Revenue
Wisconsin Department of Revenue