Am I required to file my Illinois and Wisconsin state taxes by April 15 if I'm owed a refund? (I'm aware of the federal filing requirements regarding refunds.)

Both states have penalties for filing late. The Illinois penalty charges were hard to understand. There is a link below.

Illinois states:

You will owe a late-filing penalty if you do not file a return that we can process by the due date. 

https://www2.illinois.gov/rev/research/publications/pubs/Documents/pub-103.pdf

From Wisconsin:

If you file late, without an extension, you are subject to penalties and interest. Your 2018 individual income tax return must be filed by April 15, 2019. Late-filed returns may be subject to the following charges:

• Late-filing fee -- $50
• Delinquent interest -- 1.5% per month (18% per year)
  Note: During an extension period, interest is charged at 1% per month (12% per year)
• Negligence penalty for failure to timely file -- 5% per month of the tax required to be shown on the return, up to a maximum of 25%