No the judge can't allow hims to relinquish his rights under the facts you've presented. A parent not only has rights to their children but obligations. It is against public police in Florida to let someone out of their child support obligation. What he should do is immediately file a Motion to Modify is Child Support - even if he does it him self letting the Court know about this "MATERIAL CHANGE IN CIRCUMSTANCES" since the last support order was entered. This will allow the Court to give him some relief once the situation if brought before the Court.
The answer may change significantly if there is another man involved that wishes to adopt the children. Then your husband's rights are extinguished once the adoption happens.
Agreed with previous answer. The only way he has his rights terminated is if there is an adoption.
As far as locating the children, where does the child support go? If he is supposed to pay the State of Florida, they should have an address fir the mother.
Sign up to receive a 5-part series of useful information and advice about child custody law.