There's a lot of information about Illinois unemployment insurance available here, including this brief overview (PDF) of worker's rights and this handbook (PDF). I don't see within that anything that would suggest that quitting a prior job would disqualify you for unemployment benefits, provided you qualify due to the nature of your termination from your most recent job.

There is the following statute, but I doubt it would apply based on what you have said,
Quote Quoting 820 ILCS 405/601(A) - Voluntary leaving
A. An individual shall be ineligible for benefits for the week in which he has left work voluntarily without good cause attributable to the employing unit and, thereafter, until he has become reemployed and has had earnings equal to or in excess of his current weekly benefit amount in each of four calendar weeks which are either for services in employment, or have been or will be reported pursuant to the provisions of the Federal Insurance Contributions Act by each employing unit for which such services are performed and which submits a statement certifying to that fact.
(You can read all of the Illinois worker's compensation statutes here.)

Can you share any more details about the reasons for the "questioning", even if by calling the agency and asking for the statute or other authority they are applying?