What is your question? I don't know what you mean you were fired for "hearsay," Additionally, I am unclear as to whether or not you in fact filed for unemployment. Did the unemployment office send you an official denial letter? If you were in fact denied benefits, you may appeal this finding. See the following link for more information. http://www.ides.illinois.gov/page.aspx?item=47
Please be advised that the above answer is not legal advice. Legal information is not the same as legal advice -- the application of law to an individual's specific circumstances. My "answer" to your "question" is for general reference purposes and I recommend you consider consulting a lawyer if you want professional assurance that this information, and your interpretation of it, is appropriate to your particular situation.
You have not asked a question but if you want to try to file for unemployment compensation, go ahead. Your former employer will probably object and you will have to appeal the denial. In the meantime, look for a new job. Your benefits are gone.
I don't fully understand your question but in order for an appeal to be filed, you first need to litigate the issue and get a ruling. You cannot jump straight to appeals.
The author provides the preceding information as a service to the public. Author's response, as stated above, should not be considered legal advice. An initial attorney-client conference, based upon review of all relevant facts/documents, will be necessary to provide legal advice upon which the client should then rely.