The first one is a double negative because of "No one". To avoid that use "ever" instead of "never".
I'm in doubt about the accuracy of the sentences below:
No one will never do that.
No one will ever do that.
I know that when we use nobody, nothing, etc, then the verb must be affirmative. However in these two sentences the will is affirmative, and never is just an adverb. That's why I believe that the first one isn't incorrect. And in addition, I believe "never" cannot be replaced by "ever".
Not a teacher.
It's not simply because it's a double negative that is the problem. Double negatives can be useful. But in this case, it's wrong.