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Yes, the State of Illinois can get a judgment against you, and would be able to garnish wages and bank accounts in Florida. They cannot put a lien against your home, as Florida has homestead laws in place which prevent creditors from placing a lien on your property unless it is for back taxes, a mortgage company or a mechanics lien. It would not impact her getting unemployment compensation if she were eligible in the State of Florida.
I would attempt to try and enter into a settlement or payment plan with them, its unlikely they will want to do it, but she could try. Otherwise, they would have to sue, receive a judgment and then try and collect on the judgment.