The hospital is now called Stroger hospital. (John J Stroger JR Hospital of Cook County).
Has he registered for the Illinois Adoption Registry yet? If he fills out the medical form, and lets that be shared with searching relatives, the sign up is free.
Illinois Vital Records - Adoption Records
He can also request his non identifying information be sent to him by turning in that form with his registration. I think that one of the forms will have to be notarized but they are relatively quick in their responses.
Also, if he was adopted through DCFS he may be able to get assistance from them as well.
Midwest Adoption Center
Illinois also has a confidential intermediary program available but I don't know the cost/expense that would entail.
Good luck!