The two expressions from the title, “I wouldn't ever” and “I would never”, are very similar. But are they completely equivalent or do they bear any subtle differences? If so, how do they differ in meaning, usage, maybe emotional load? Or perhaps one of them feels "more natural" than the other?
Both phrases are grammatically correct and carry the same emotional connotation.
The only slight variation between the two is the contraction used in the phrase "I wouldn't ever" which makes it a little less formal than "I would never".
When writing formally, it is considered unsophisticated to use contractions (i.e. wouldn't instead of would not). Therefore, the phrase "I would never" is favorable in that instance.
But if formality doesn't matter, then both are correct. It simply comes down to preference.
I wouldn't ever = I would not ever.
They are, in fact, identical in meaning. To my ear -- I am a native speaker -- they are identical in meaning, connotation and mood.