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Both "either" and "neither" are singular in form. They should only be viewed as singular when they are used as adjectives.
Either option was considered to be an improvement.
Neither man was chosen.
When "either" or "neither" is used as a pronoun, this same sense of being singualr should continue. However, when the pronoun is followed by a prepositional phrase with a plural object, some speakers/writers will switch to a plural verb. This practice is more common with "neither" for some reason.
Either of the options was considered to be an improvement.
Neither of the men was chosen.
Some will use "Neither of the men were chosen" because there are two men. But "neither" treats the two men as one or the other, not both.
These usage notes are from the AHD:
USAGE NOTE The traditional rule holds that either should be used only to refer to one of two items and that any is required when more than two items are involved: Any (not either) of the three opposition candidates still in the race would make a better president than the incumbent. But reputable writers have often violated this rule, and in any case it applies only to the use of either as a pronoun or an adjective. When either is used as a conjunction, no paraphrase with any is available, and so either is unexceptionable even when it applies to more than two clauses: Either the union will make a counteroffer or the original bid will be refused by the board or the deal will go ahead as scheduled.•In either … or constructions, the two conjunctions should be followed by parallel elements. The following is regarded as incorrect: You may either have the ring or the bracelet (properly, You may have either the ring or the bracelet). The following is also incorrect: She can take either the examination offered to all applicants or ask for a personal interview (properly, She can either take … ).•When used as a pronoun, either is singular and takes a singular verb: The two left-wing parties disagree with each other more than either does (not do) with the Right. When followed by of and a plural noun, either is often used with a plural verb: Either of the parties have enough support to form a government. But this usage is widely regarded as incorrect; in an earlier survey it was rejected by 92 percent of the Usage Panel.•When all the elements in an either … or construction (or a neither … nor construction) used as the subject of a sentence are singular, the verb is singular: Either Eve or Herb has been invited. Analogously, when all the elements in the either … or construction are plural, the verb is plural too: Either the Clarks or the Kays have been invited. When the construction mixes singular and plural elements, however, there is some confusion as to which form the verb should take. It has sometimes been suggested that the verb should agree with whichever noun phrase is closest to it; thus one would write Either Eve or the Kays have been invited, but Either the Kays or Eve has been invited. This pattern is accepted by 54 percent of the Usage Panel. Others have maintained that the construction is fundamentally inconsistent whichever number is assigned to the verb and that such sentences should be rewritten accordingly. See Usage Notes at every, neither, or1, they.
USAGE NOTE According to the traditional rule, neither is used only to mean “not one or the other of two.” To refer to “none of several,” none is preferred: None (not neither) of the three opposition candidates would make a better president than the incumbent.•The traditional rule also holds that neither is grammatically singular: Neither candidate is having an easy time with the press. However, it is often used with a plural verb, especially when followed by of and a plural: Neither of the candidates are really expressing their own views.•As a conjunction neither is properly followed by nor, not or, in formal style: Neither prayers nor curses (not or curses) did any good. See Usage Notes at either, every, he1, none, nor1, or1.