No. What she's doing is technically called "extortion" and it's illegal, but it's child support modification request, so it's pretty lame as "threats" go.
As for CS going up...that's a possibility if it has been more than two year or your income has gone up (at least a buck an hour), but then, it might go down too.
How do we ensure that he keeps his rights?
The ONLY way your husband's parental rights can be forcibly taken away is for him to abuse, neglect and/or place his children in danger. Mind you, the State does NOT lightly terminate parental rights...you have to be a REALLY poor excuse for a human being before they will do that.