The following language is repeated many times in the Bright Start Program Disclosure Statement & Participant Agreement: "Individuals who file individual Illinois state income tax returns may deduct...". I did not see any language akin to "Illinois residents" when describing the deduction. So, based on this, it appears that you can take an IL deduction for contributions into an IL 529.
To be safe, I sugest you read the Disclosure Statement for yourself. Also, thoroughly check the IL-1040 instructions in the part about 529 deductions and see if it is limited to residents only. Finally, as I am no accountant, you should check with yours, if you have one.