Can Esurance deny a refund if I forgot to cancel my policy when I switched to a new company?

I switched companies in December 2016. I kept getting auto billed every month on the 20th from Esurance (I did not notice -- my oversight). When I requested a refund because I was insured with two companies at the same time, they told me, "Unfortunately in the sate of Illinois we are only able to backdate the cancellation 30 days from your call." Verbatim from their email. I provided my new policy paperwork showing the start date of 12/23 with the new company showing I was never without insurance. Is this a state law?

This sounds like an company issue and not a state law. Insurance companies in Illinois are legally required to issue refunds for "unearned premium", but you didn't prepay your bill. Long story short, they only have to repay the portion of your premium that extends past the cancellation date; i.e. you paid your bill on the 1st and cancelled on the 5th, then they would refund you from the 5th through the end of the month. Unfortunately for backdated cancellations, refunds are usually up to the company. You might consider calling Esurance and ask to speak with a supervisor. If that doesn't work, contacting your state's insurance department about the issue or even an attorney may be the only other option.