The father of my boys now all of a sudden does not believe they are his but refuses to take a DNA test and has threatened all sorts of legal action (I am sure to make everything dramatic and tie me up in court to spite me). Since now he does not want them anyway and he is not on the birth certificate can he sign away his rights?

He cannot sign over rights regardless if he is on the BC or not. Unless someone is going to adopt them, there is next to no chance of rights being terminated.

If you want to, take him to court and have paternity established. His refusal to take the DNA test becomes an order and he can take the test or go to jail.