I live in Florida but work for an Illinois based company. Since I don't file state taxes in Florida, must I file Illinois state tax?

No.   If you live and work in Florida, you do not have to pay IL income tax just because your employer is located in IL.  FL has no state income tax.  You simply file a federal tax return.

Q. Must I file Illinois state tax?

A. No, unless you sometimes physically work in IL (e.g. business meetings or training at the company office).