In the case of 'Appreciation on sale of employer securities' for IL state tax return, does this option apply to the sale of employer stocks that were acquired as RSUs?

Illinois has a separate form (form IL-4644) that needs to be filled out to cover the appreciation in value of employer securities.

 

There isn't an extra tax break.  In fact this form will make TurboTax require you to mail the return.

 

Here is a more detailed thread on the IL-4644.  There is some discussion as to whether the form applies to RSUs (although I believe that it does).

 

@ghaemi-kasra