"Force men to give up their DNA for testing?"
Of course. Why not? Court ordered DNA testing isn't illegal.
Per se ~
"Can a Potential Father Refuse to Take a Paternity Test?
In many states, an action for determining paternity is actually considered to be a civil lawsuit. In such claims, a court will often order the alleged father to submit to a paternity test at a court-approved facility. Court determinations of paternity are final- once the test has been established, a judge will make a final decision as to the rights of the child and parents.
If the person refuses to take a court-ordered paternity test, it could result in legal consequences. For example, the potential father may be held in contempt of court for violating the court order. This can result in fines or criminal charges. Or, a judge may enter an automatic (default) judgment against the person."
If a man is determined to be chronically and criminally negligent, it wouldn't be much of a stretch to order a forced sterilization on him any more than it would be to order it for a woman.