"Both" in a negative sentence

Anonymous"Both of them are not showing up today"Is it incorrect to say this?

It's correct to me.

Anonymous"Neither of them are is showing up today"?

"Are" is not used in formal English.
I would also prefer your latter sentence to the former one.

Use a positive statement:
Both of them are going to be absent today.
Both of them were absent today.

http://dictionary.cambridge.org/us/grammar/british-grammar/linking-words-and-expressions/both

In a negative statement,

AnonymousIs it incorrect to say this?

It's not grammatically incorrect, but it makes the logic ambiguous whenever you have a universal quantifier (all, both) in the subject position followed by a negated verb.

All of them are not here can mean Not all of them are here (~ Only some of them are here) or None of them are he

AlpheccaStarsNeither of them have showed up today.

I would use "has" here. Would you please teach me the way you considered it correct?

I tend to favor proximal agreement when there is a prepositional phrase (of + plural noun or pronoun) after "neither. When there is no prepositional phrase, I use the singular form.

Both of them dont like the idea