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Dear Ms. Bieberich,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located 71 series for Illinois and 34 series for Ohio in the Records of the Provost Marshal General's Bureau (Civil War) (Record Group 110) that pertain to the draft and enlistment of soldiers. They may include the information you seek about enlistment requirements. For access to these series, please contact the National Archives at Chicago (RM-CH) via email at firstname.lastname@example.org.
In addition, there is a Prologue article titled Civil War Draft Records: Exemptions and Enrollments that may be of interest.
We hope this is helpful. Best of luck with your family research!