I always pronounce “either” as EYE-ther, and “neither” usually gets the same pronunciation. But I think I’m more likely to slip in a “neether” once in a while.
I actually grew up saying “eeether” but when I was in high school I got to know some people who said “EYEther” and thought to myself, “gee, that sounds so much nicer” and started training myself to pronounce it differently.
Yes, that was exceedingly pretentious of me, but hey, I was a teenager. Anyway, my adopted pronunciation eventually became my natural one. I’ve been saying "EYEther for so long that it would take a huge effort to switch.
I went through the same process with crossing my sevens, at about the same time. After decades of seven-crossing, it is totally natural to me. But when I started, it was a bit of an affectation.
UK…
How odd, I’ve never actually thought about it before but I say both EEEther and EYEther. (Especially when saying either/either as EEEther/EYEther, which roughly translates to “I don’t mind either way, it makes no difference to me”)
I just seem to switch between the two at random. I’d probably say “EEEther way”, but also say “it can be EYEther”.
I’d always say NEYEther though, and would only say NEEther when saying “either/either, neither/neither” which is an extended form of the either/either idiom as I know it.