I thought I'd add something to what has already been said in @PLL's answer. Merriam Webster's Dictionary of English Usage, which I like for its descriptive style and useful examples, has this to say about neither:
The reason it is sometimes plural is easy to see when you think about it. Neither serves as the negative counterpart of either, which is usually singular. But it also serves in the same way for both, which is usually plural.
The reason it seems more natural to use the plural verb to negate two choices is that we want to negate both choices, and not just one of them. For example, it may seem more natural to say, "Neither of these colours suit me" and "Neither yellow nor orange suits me".
It also says:
The singular number of neither is most likely to be ignored when it is followed by of and a plural noun or pronoun, for then both notional agreement and the principle of proximity pull in the direction of a plural verb.
It gives numerous examples of neither taking a plural verb:
- Thersite's body is as good as Ajax'/When neither are alive--Shakespeare, Cymbeline, 1610
- Neither belong to this Saxon's company--Sir Walter Scott, Ivanhoe, 1819
- He had two job offers, but neither were ones he felt he could accept--Diana Diamond, N.Y. Times, 1974
In conclusion, it says this:
The pronoun neither, then, is not invariably singular, though it is more often so. When formal agreement obtains, it takes a singular verb. When notional agreement obtains, it takes either a singular or plural verb. These constructions are neither nonstandard or [sic] erroneous. If you are writing something in a highly formal style, you will probably want to use formal agreement throughout. Otherwise, follow your own inclination in choosing singular of plural constructions after neither.