The nuisance wildlife control operator should euthanize any muskrats that are trapped if they plan to use live traps to capture them. To release wildlife (including muskrats) onto another property, one must have the written permission of that landowner. I doubt there will be a landowner willing to accept the muskrats. It is illegal to release them into state parks, natural areas, or parks. Those areas with suitable habitat already have populations of muskrats.
Muskrats are best trapped in their runs under the water. This prevents accidental take of non-target species. The traps normally used to trap muskrats include the No. 110 Conibear® traps or No. 1, 1½, or 2 leghold traps.
Another option is habitat modification. Depending on what sort of damage the muskrats are causing, an Illinois Department of Natural Resources wildlife biologist may be able to give you suggestions on ways to both protect the shoreline and provide habitat for the muskrats. In Lake County your contact would be: Stefanie Fitzsimons (815) 675-2386 ext. 215
In Cook County your contacts would be: Joe Rogus/ Bob Massey (815) 476-2523