According to Strunk & White's The Elements of Style:
With none, use the singular verb when the word means "no one" or "not one."
None of us is perfect.
A plural verb is commonly used when none suggests more than one thing or person.
None are so fallible as those who are sure they're right.
Any reason beyond the simple rationale given here is, IMHO, superfluous. As for the singular use of neither, Strunk & White group this word with its counterpart either and others:
Use a singular verb form after each, either, everyone, everybody, neither, nobody, someone.
Although both clocks strike cheerfully, neither keeps good time.
The logic here is easy to see if you consider that both either and neither conceptually separate two items rather than group them together, much like words such as everybody [i.e., each individual body] and nobody [i.e., not one body] do.