They can't take his citizenship...
Since he claims to be a born citizen, he has citizenship by birthright and nothing CBP can do can possibly revoke it.
He can voluntarily renounce his citizenship, but he has to do that through the State Dept. (which CBP is not part of). And that is an elaborate and expensive process that can't even be done inside the United States. If someone could do it merely by entering without papers and asking for a self-deport, lots of expats would save a lot of money - and that's not gonna happen :)
...but they could put him to serious inconvenience
In this particular case, CBP found his documents suspect. Probably because (if it's the case we've seen documented elsewhere) he was with two other people whose entry was illegal, and they had forged documents.
So most likely, if he agreed to self-deport, CBP would use that as prima-facie evidence that he is not a bona-fide citizen, and therefore, that his papers are faked. They certainly will not give fake papers back to someone who has tried to pass them.
So the victim would be obliged to go back to SSA, the state, etc. and re-acquire his identity documents. From outside the country. It's a pretty big chore.