I got curious and checked Illinois' hunting regs on coyotes. I couldn't find any restriction to shotgun or .22Lr for coyotes on public lands BUT the DNR regs did say to check with the "site Office" for any specific regulations for that site only. If you google search, you can find some guys on different hunting blogs saying they hunt coyotes on public lands with 243's and 308s so the restriction that DNR warden told you about may only apply to a specific public land "site." You should check out a few of the site offices for different public lands and see which ones don't restrict you to shotgun or 22LR. Good luck.