The negative polarity particle *ne is reconstructed for Proto-Indo-European from "no" forms in pretty much all Indo-European languages.
*aiwo < *h2eyu-, *h2ey- had, on the face of it, no relation with *ne. It's not as well attested, I believe. aye is probably a loan from Norse into English, anyway not just a variant of yea vel sim., as one might be predisposed to think.
Direct continuant in English seems to be ǣ "ever". age also belongs here if I remember correctly.
This is the basic distinction the author is trying to draw but the verbiage is confusingly ambiguous. Specifically in the following two excerpts from the two quotes, respectively
... *gikhwatharaz, ancestor of English whether
... hwæther ‘which of two’ (ancestor of modern English whether).
Since, furthermore, "the compound ǣgehwæther" shows the element *gi- reflected in -ge-, it is heavily implied that wether < hwæther stemmed from a mere *hwatharaz. This is not ambiguous in total, if the entry for whether should resolve this.
It is ambiguous though, because it might be implied (knowingly or unknowingly) that *gi(h)- was perhaps dropped or lenited before hwæther. That might be a reasonable assumption if comparing *wiþrą, *wi- (cf. wither), while respecting the difficult development of velars in Germanic (point in case, I'm pretty sure the g's should be dotted, read /y/).
But Wiktionary confirms a wh-word from *hw < *kw. Quod Erat Demonstrandum.
The problem, to me, is that they don't clearly explain *gi(h)-. This could pose an interesting question, but that would lead far too far into deep open waters.
In summary it suggests itself:
Which does effectively, in the end, look a lot as if n- was simply tacked onto (ei)ther. They touch not upon the vowel qualities, whether diphtongized or long, that may have influenced each other, the spelling, and maybe the usage.
So what he's really saying is that *gikhwatharaz had not developed to be homophone first with hwæther "*either" and second to merge with ǣgehwæther in ǣgther, which may have existed in two variants with and without dot.
I dare say he might be wrong in some respect. It's imaginable that anumber of particles reconstructed for PIE had n-infix and that these were reconstructed summarily as one. n-infix isn't well understood and othr ways supposed to be excrescent nasalization in Germanic (so cold up north, *sniff*), but the fact that *new "new" is quite the opposite of ae "ever" where *u in the reconstruction represents a vocalized semi-vowel *w pretty much speaks for itself; especially if never! "no, nah, nu'uh" stands to reason, while ever has no commonly accepted etymology that I'm aware of. The fact that English neither has no clear cognates does not yet mean it was definitely created in Old English. It only means there is no external evidence to the opposite. Modern German rather says entweder ... oder or weder ... noch (OHG noh)