It's not true that Burmese do not have any natural predators in the Everglades. They have MANY predators, from raccoons and opossums to herons and raptors when they are small, all the way up to adult alligators once they are fully grown. No doubt a black bear would figure out a way to take advantage of this new prey source as well.
Yes, Burms eat a lot of animals--rodents and baby alligators, young birds, etc--but they're also eaten by a lot of animals, particularly when small. This isn't a case where a species is introduced to a place where it has no natural predators. MANY animals in the Everglades eat snakes, and the burm is not so significantly different that it's not on the menu. We KNOW alligators are eating them. It's much too early to declare that Burmese are going to cause harm to the everglades ecosystem.