I use a 12 gauge all the time, just use a light load, like #6 or 7 1/2, in a light game or even target loads.
If you are talking about this year, Marseilles is closed for the season (they open on September 1), Starved Rock is closed, and they only allow a short squirrel season on the south end of Matheissen in (I think) August and September. I don't know much about those sights toward the south of Cook, but I know most public sites around here close their squirrel seasons by about October 1 to make room for archery hunting (Marseilles is an exception). So, as far as Kanakee or DesPlaines, I don't know, but I doubt it. I've heard Mazonia has a late upland season, you might want to call there. The only ones I'm sure are still open this year are Marshall, which is 45 minutes south-west of Starved Rock and Big Bend, which is out on the Rock River (quite a drive).