Are Liscenced Clinical Professional Counselors in the state of
Illinois permitted to diagnose mental illness?

Diagnose means: 
"To determine the existence of a disease or condition or to
distinguish one disease or condition from another."

(just one of the similar choices provided by Google   define:diagnose)
Diagnosis is not treatment.  A layperson can diagnose  - in a loose
sense of the word -  many diseases, physical as well as mental.
It would seem that a clinical professional counselor should have the
knowledge and experience to recognize signs of mental illness and
would be negligent in not recommending that the person seek
appropriate professional care.