There are a fair number of factors here. I'm not sure how messing up your FAFSA stuck you with a bill. FAFSA gets you aid, unless you mean that your screw up resulted in less aid than you anticipated. Even still, I'm not sure a state university has the right to intercept your state taxes, but that might be a particular provision of Illinois law which I'm not familiar with. If this is a tuition debt being collected by the university as you say, your federal taxes should be safe. The only way your federal taxes can be touched is through the Federal Dept. of Ed. for a default federal loan.
I suggest you find a consumer attorney in your state who can verify the facts and the particular state law that seems to be costing you your state taxes.