I’ve been known to pronounce it either way.
It’s “EYE-gor.” 
I mostly say “EEE-ther,” but every once a while I’ll say “EYE-ther.”
eyether - but I’m pretty sure that it was an affectation that stuck.
But then you must part.
Either is correct; neither is wrong. You’re welcome. 
I knew someone would beat me to it.
That’s me! The last sentence, I mean. And the amount of switching seems inversely proportional. Off the top of my head, I’d say my usage is
Either – 70% EEE-ther | 30% EYE-ther
Neither – 70% NYE-ther | 30% NEE-ther
I have no strong preference - I use each about equally.
(I also use both “American” and “European” modes of knife-and-fork.
Raised in Midwest. Escaped at age 30.
I use both, inconsistently- England.
Southeastern US. EE-ther/NEEther. I think I could probably count the number of times on one hand I’ve heard the other pronunciations used (outside of a TV or movie).
On this, I go either way, depending on… I’m not really sure.
EEEther, nEEEther.
I get the sense that nEYEther tends to come from people who pronounce aunt ont instead of identically to ant.
British – family originally from north-west England. Always EYE-ther and NYE-ther. I think that in Scotland, people more often say EE-ther / NEE-ther; but couldn’t totally swear to that.
Oh fuck. I just read that in my head as “eether is correct, nyther is wrong.” Now I’m totally schizophrenic. Thanks.
EE-ther and NEE-ther. San Francisco Bay Area.
Wow, a lot of interesting responses. I think, like some have said, I use EEE-ther most of the time but EYE-ther sometimes for emphasis. As in:
Son: I don’t know whether I want to do my homework before or after dinner.
Me: EYE-ther way, you’re getting it done!
Either – 40% EEE-ther | 60% EYE-ther
Neither – 90% NYE-ther | 10% NEE-ther
As I’ve gotten older I think my EYE-ther percentage has climbed. The EYE sound goes better with ponderous and pedantic speaking. Something you all know I’m (too) good at.
You say “Hey, you wanna Bud or a Corona?” I say “Eee-ther.”
You say “Would you prefer the vichyssoise or the salade nicoise?” I say “EYE-ther would be lovely, thank you.”
It varies depending on the context, but I usually prefer “EEE-ther.”
I’m from Lawn Guyland, Noo Yawk (all my life).
Abbott & Costello used “EEE-ther.”
I pronounce them correctly, of course. 
I think I use nye-ther more often. I have no idea whether I use eether or eyether, except that I always say “eether-or proposition”.
I pronounce Aunt as Are-nt and I pronounce Neither and Either both ways totally at random.