I have my brother's children (twin 22 yo's) living with me. She has a disability (ADHD) and receives money as a survivor of a veteran, and her brother is unemployed. Their father passed away in May of 2010, and they had nowhere to go. The girl has been helping out with paying towards the utilities, but I have never asked for rent of any kind. She has become a real problem child, refuses to help or pay toward the utilities, and I want her to leave but she refuses. Do I have any recourse to evict her under these circumstances?

Sure. If the person is not a listed owner or lessee (renter) you can evict them. To do it properly in Illinois you may need to serve her with a notice to evict as a Court may find you had some type of month to month "lease" with the payments for utilities. She can't squat there. Contact a landlord-tenant attorney to proceed with an eviction.