I currently live in Texas but I am considering a full-time work-from-home position with a company from Illinois. Will I have to file Illinois state income tax?

I don't know specifically about Illinois, but as a general rule, you pay taxes (1) in your home state and (2) in the States that you actually perform work in.  If Illinois follows that 'common' rule, you would only pay tax on the income that earned while physically in Illinois.

However, you may want to discuss with the HR department how they will be handling it.

It's total legalese, but here is the actual Illinois statute regarding non-resident income taxable by Illinois:

Thanks @TomD8

According to the Illinois DOR:

As a nonresident, you must file Form IL-1040 and Schedule NR if

• you earned enough taxable income from Illinois sources to have a tax liability (i.e., your Illinois base income from Schedule NR, Step 5, Line 46, is greater than your Illinois exemption allowance on Schedule NR, Step 5, Line 50), or
• you want a refund of any Illinois Income Tax withheld in error. You must attach a letter of explanation from your employer.

http://www.revenue.state.il.us/Individuals/FilingRequirements/

If your business trips are for a "a temporary or transitory purpose," you are still considered a nonresident.

From the Schedule NR instructions:  If you were not a resident of Iowa, Kentucky, Michigan, or Wisconsin, you must attach a statement from your employer, on company letterhead, indicating that your wages were not earned in Illinois and Illinois tax was withheld in error.

It appears your employer should not withhold taxes for the income earned while in Texas.