Prepositional phrase vs. subject

Since "either" is singular, it should technically be "does". However, native speakers will often use "do" (especially in conversation) and many people would not perceive this as an error.

I think the use of "you" makes anything other than "do" simply sound awful!

Hmm... It certainly doesn't sound awful to me -- just more formal, or as if the writer/speaker is consciously trying to be grammatically correct. Google Book Search prefers "do", but not overwhelmingly:

633 on "does either of you"

958 on "do either of you"

You're right. On further reflection, I'm pretty sure I've said over my shoulder to the children in the backseat "Does either of you have to go to the bathroom? Tell me now, because we're not stopping again!" as well as "Does either of you know how much trouble you will BOTH be in if this doesn't stop this instant?!" (Although the first was a real question, the second was rhetorical.)

I

Referring to Paul Brians, as a subject "either" is singular ("Either ketchup or mustard is good on a hotdog.") However, if "either" is modifying a subject in an "either ... or" phrase, then the number of the verb should be determined by the number of the second noun: "Either the puppy or the twins seem to need my attention every other minute."

I would say it was a question of focus:

1. Do either of you have a match?

— "either of" modifies "you": focus on the latter.

2. Does either of you have a match?

— "either" is modified by "of you": focus on the former.

MrP

victor_amelkin"Either the puppy or the twins seem to need my attention every other minute."

This is not the same. The sentence posted by the OP is a question and the verb that denotes time and number, "do", comes before "either", not after the nouns.