Of course, “He never eats breakfast” is grammatically correct. – although I always think that it’s better to eat breakfast 😉

never, usually, often etc are one word adverbs of time/frequency which are used before the main verb:

-She usually has cereals for breakfast.
-The team often play on Sundays.
-John has always lived in Birmingham.
…. but after the verb “be”:

-Fred is always happy.
-Miriam was usually a very good student.

For more information about adverbs you can start here:
https://learnenglishlanguagewell.com/2015/how-to-use-adverbs1/ ‎

We put “es” after (x-o-s-ss-sh-ch-zz) in the third person singular because of its pronunciation. With spelling in English (and probably other languages) we have to consider that the spoken language existed before reading and writing was established. Therefore the spelling of the words that end in sibilant sounds (x, s, ch, z, sh) reflects how we pronounce the words. This is because if we want to pronounce these verbs – miss, mix, match, mash with an additional “s” sound we need to add a vowel sound after the x, ss, ch, etc or it becomes extremely difficult to pronounce. So the spelling: misses, mixes, matches and mashes guide us to the correct prounciation for the written word.
Also with verbs that finish with “ss” (miss, kiss) if we only put an “s” and not “es” would we spell them with 3 “s”: kisss, misss ?

I will write an article about this and then I’ll send you the link. OK?