It would help to have a clear picture of what happened here, but I'll imagine that California residents had income from an Illinois business and/or real estate rental even before they moved to Illinois. While they lived in California, they had to pay tax there on that income.  So they qualify for Schedule S -- as that form's instructions state,

"California part-year residents:

• Follow the instructions for residents for the part of the year that you were a California resident."

And the credit can be shown on Line 58 or Line 59 of the 540NR.  If the software doesn't compute it automatically, you won't be the first preparer who has had to do it manually.  

Or did the Illinois income not start, until they left California?  Then don't show it on the California return in the first place.