I would like to make sure I understood the usage of these:
Do you want A or B?
- I do not want either. [none of them]
- I want neither. [Can I say that?]
are both fine.
Neither is the negative form of either.
There are several ways you could express your sentiments:
But I would not use:
(That's what neither is for.)
If, after declining A and B, you were then offered C, D, or E, you could say:
("I don't want either..." sounds more acceptable with more than two elements, as opposed to just two elements, which is why I opted to include it among the possibilities here, but recommended against its use in the previous example.)
Still, you would not say:
because that forms a double negative, which is not used in Standard English.
Lastly, as Martin B pointed out, the word either has two meanings. It can be used either as a word that introduces two alternatives, or as a word that indicates similarities with a statement just made. So, in the scenario just described, you could also say:
but in those four cases, either is being used in its other context, and is roughly equivalent also or as well.
You can say either
Either indicates a choice between two alternatives. Neither combines two negative ideas.
Often with neither you repeat the options, joining them with nor.