It's such a bad problem here in New England that I can't even google to find out which worms ARE native.
I suspect this has to do with the nature of the word "native." Nearly all earthworms in North America (most certainly New England) are non-native, brought over from Europe. And it's not so much that European worms were aggressive and out-competed the native worms. It's that the ice age wiped out all the worms and left the forests without any. It's possible there were worms before the ice age. It's possible the boreal forests looked different then, too. It's possible all of this is a natural cycle and that the re-introduction of earthworms and the nutrient cycle that accompanies them was inevitable. It's possible that we have introduced an extraordinarily invasive species that will kill our boreal forests. We don't really know. I've read studies claiming invasive worm species are bad, I've read studies claiming the same worm species as rehabilitating.
However, I can tell you with certainty two things about red wigglers (Eisenia fetida), the most popular composting worm:
1. These worms are surface-dwelling, manure eaters. They do not thrive in the forest. They require high nitrogen inputs and generally do not spread very far outside composting piles. Freezing temperatures kill them as well (though not their cocoons).
2. Their pace of breeding far outstrips any efforts mere humans could enact to reduce their total numbers. A mature red wiggler can produce two to three cocoons per week. Each cocoon averages three hatchlings. Each worm is a hermaphrodite and can fulfill whichever sex is necessary to reproduce.
All the most popular composting worms (Red Wigglers, Blue Worms, European Nightcrawlers, African Nightcrawlers) are already established throughout America, and I doubt there's anything we can do to change that. The USDA considers Dandelion an invasive species as well, but you don't see many people worrying about it. I consider worms to be similar. These introductions all happened in the 18th century, and there's just not much to be done about it at this point.