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?]
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There are several ways you could express your sentiments:
I don't want either. I don't want either of them. I want neither. I want neither of them. I want neither A nor B.
But I would not use:
I don't want either A or B.
(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 C, D, or E. I don't want any of those. I want none of those options. I want neither C, nor D, nor E. I don't want either C, D, or E.
("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:
I don't want neither C, D, nor E.
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:
I don't want C, D, or E, either. I don't want any of those, either. I want none of those options, either. None of those options appeal to me, either.
but in those four cases, either is being used in its other context, and is roughly equivalent also or as well.