May I take the K-12 Education Expense Credit if I was a nonresident of Illinois?

No. You may not figure a credit for qualified education expenses unless you and your student were Illinois residents when you paid the expenses. See the  Publication 112, Education Expense Credit;   Publication 119, Education Expense Credit General Rules and Requirements for Home Schools; or  Publication 132, Education Expense Credit General Rules and Requirements for Parents and Guardians, for more information.