Generally they will not enable one to overturn a federal criminal conviction. If the individual is charged with illegal entry or illegal reentry after deportation, if he can prove derivative citizenship it can overturn a conviction. The individual should file a Habeas Corpus motion.
Immigration has its own set of rules and records. The errors you may have found on an Immigration file could be clerk input. You will need an attorney who handles both immigration and criminal law in the Federal system in your location to review and attempt to challenge any disputes between the two records.