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Forms for U.S. Green Card (I-485) & Citizenship (N-400) explicitly ask under oath if you have ever worked in the U.S. without authorization. If you answer "No" to conceal unauthorized J-1 work, you have committed material misrepresentation and could be denaturalized.

The answer is "yes", and his status was regularized under 245(i). Fedorenko v. United States (1981): citizenship is only "illegally procured" if the underlying Green Card was granted contrary to statutory law. 245(i) legally grants a green card. Also, unauthorized work isn't a citizenship bar...