I paid a retainer fee, but did not sign a contract, now the attorney is stating I did not hired him for the case. What right do I have for this particular situation?
Your question involves the professional responsibilities of the lawyer in question. Generally speaking, an Illinois attorney's conduct is governed by rules made by the Illinois Supreme Court. Rule 1.5(a) requires a lawyer's fee to be reasonable, and Rule 1.5(b) requires the lawyer to communicate the basis of the fee to the client within a reasonable time after commencing representation.
Note that there is no requirement in Illinois to have a written arrangement with an attorney unless the attorney was planning on charging you a contingency fee (meaning you pay a percentage if you win). However here, it appears (based on your facts) that the lawyer did not meet Rule 1.5(b) and did not follow up with you on what rates he / she was going to charge. This may be the cause of the confusion in the lawyer's office. There are additional professional requirements a lawyer must meet pertaining to retainers, depending on the circumstances of your arrangement.
I would suggest sending a letter and a photocopy of proof that you paid the lawyer (cancelled check, debit or credit card statement, etc.), after it is properly redacted to keep personal information safe, to an organization called the Illinois Attorney Registration and Disciplinary Commission (IARDC). They investigate and recommend discipline for Illinois attorneys. They can look into the matter and help you figure out what happened. The ARDC's website is www.iardc.org.
In any event, if a lawyer (or anyone) took your money and did not provide goods or services you have a legal right to have the money returned. If the ARDC option does not sound appealing, an attorney will be able to assist you further.