Why in the WORLD do you think he should get out of supporting his child--if it is his child? Do you think that just because he's not in a relationship with the mother that it means he gets to just walk away from supporting the child?
It's really too bad that so many people think the way that you do. Maybe if more people actually thought that people SHOULD support their own children, there would be fewer children born of one night stands because people would actually THINK before they dropped their pants.
Legally, your husband is responsible for the child if he's the father. He should definitely get a DNA test, but if the child IS his, the state WILL come after him for the support they are providing the mother.