Jen,
If I recall correctly, this isn't the first time your son mentioned giving up his rights. Unfortunately, it doesn't work that way. The only way he could do it is if his ex's new boyfriend wants to adopt them. Then they could file the paperwork to have his rights terminated. They would have to go to court and show that it would be in the best interest of the children, and that your son doesn't object. You can't just sign your kids away because you don't want to pay support.
I also recall that your son has a long history of not paying and then coming up with the money when they threaten jail. It doesn't seem unusual that they would at some point get tired of this, regardless of the fact that he's working and now has another child to support. I can only imagine how many times he's been warned. From my personal experience, running away won't solve the problem. they may lose track of him for a while, but when they catch up with him what he owes will be retroactive. My brother spent over 10 years paying back child support for a son he hadn't seen since he was 3. The state his ex moved to sent him a letter when the child was a teenager letting him know how much he owed.
Genny