My permanent home is in CA, while I was in school in IL. I earned money in both. TurboTax insists on classifying me as NonResident of CA and Resident of IL.

Your state of residence is where your true, fixed and permanent home is and where you intend to return.  Based on this Illinois Department of Revenue website, you would be considered a non-resident of Illinois:  

"Nonresident – You are a full-year resident of a state other than Illinois.

Example: Rob, who is from Missouri, attends college in Illinois (and pays nonresident tuition). Rob is a full-year nonresident, even though he lives in Illinois for nine months of the year while attending school."

In the My Info section of TurboTax, did you select California as your state of residence?  Even if your mailing address that you entered in TurboTax may be in Illinois, you can select your actual state of residence on the Your info page as California.  If you do that, your Illinois return will be a non-resident return and your California return will be a resident return.

When you do go to prepare your state tax returns, make sure to prepare the Illinois non-resident return first.  If you already started both state returns, you may want to delete them and start over so that the correct information gets carried through on them.  To delete state returns in TurboTax Online:

1. Click on State on the left side menu, then click Go to state return in the lower right corner
2. Click the trashcan icon to the right of the state return, then click Delete on the pop-up to confirm deletion.  Repeat this for the 2nd state return.
3. Click on +Add another State, select a state from the dropdown, then click Continue.  Repeat this for the 2nd state.

In Turbo Tax Desktop:

1. Click the State Taxes tab, then Continue
2. Click File in the black banner on the left and select Remove State Return from the dropdown
3. Highlight a state return, then click Remove.  Repeat that for the 2nd state return.
4. The states will still be present on the Your State Returns page, but all the previous data are cleared so that you can start over on them.