Can my Name be kept off my Divorce Documents in the Public Record in Illinois?

Probably not. A judge would  have to find "good cause" and it almost never happens unless you are famous. That being said, be aware that in order for someone to find your divorce papers, they would have to know that you were getting divorced, where you got divorced- and then go through the trouble to pull your file.... 

Carol O'Connor Cadiz
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Attorney & Owner at O'Connor Cadiz Law: Bankruptcy, Injury, Real Estate & Mediation