Can I receive the Illinois tax credit for education expenses (private school tuition) I spent on my child who I am not claiming as a dependent?

Yes, but you and the child’s other parent can’t claim the same expenses.

According to an FAQ on the Illinois Revenue website, you can claim the credit for a child you don’t claim as a dependent on your tax return

“… if you are the parent or legal guardian of an eligible student and you paid qualifying educational expenses in excess of $250 to a public or nonpublic school in Illinois. However, if your child's other parent or legal guardian files a separate return and claims the K-12 education expense credit, you may not both claim a credit for the same expenses and the total amount of credit claimed between both parents or guardians may not exceed $750.”

See also the Illinois Department of Revenue’s  Publication 132, Education Expense Credit General Rules and Requirements for Parents and Guardians for more information.

Can one parent claim as depends and the other claim college expenses!? The parent claims as dependent makes too much money to claim college expenses..