Should this be a federal crime?

If you check a box on an official government form saying you are African American, thinking since we all came from Africa at some point, then it applies, and you do so knowing full well that you will receive race based benefits, should you go to jail if others classify you as white?

How would such a law be worded and enforced?

Yes, it should be a federal crime and the first person to get arrested would be this fraud and grifter: Jeffrey "Shaun" King.

Absurd ethics leads to absurd rules, absurd rules lead to absurd scenarios.

How exactly would you enforce this? Would you need a court ordered DNA test?

Yes, either a DNA test or other documented ancestry proving it.