Look, I get the need for grammatical correctness. I do. But there are times in which the double-negative more aptly describes the sentiment being expressed. Yes, technically, the correct version would do the same For example.
I don’t not want pancakes.
I want pancakes.
In both instances, the subject is expressing a desire for pancakes. But the difference is that one is denying a state of disapproval for something, while the other expressing a distinct desire for said pancakes.
So when LeBron says “I’m not going to not take myself against anybody,” that’s a lot different from “I will take myself over anyone.” One is a statement of belief that you would take whoever you were playing one-on-one. The other is an expression that you’re not going to short yourself or go in lacking belief in yourself. It’s just more accurate for him to say that. The mindset is “No one can beat me,” versus “I can beat anyone.” It’s subtle, but it’s there.
Oh, and by the way, maybe not ask him the question anymore. Yeesh, Lebron. Chill out, Mr. Man.