Me neither or me either? What is correct?

Short answer -- in everyday informal speech "Me neither" is often the best choice.

It is true that for FORMAL usage, you would be expected to say something like "Neither do I" or "Nor I".

But for INFORMAL/COLLOQUIAL --everyday!-- spoken English, those forms do not always need to be used, and may even be discouraged as "stuffy".

To start with, it does not matter at all that it is a "sentence fragment". Simple positive or negative responses are commonly 'incomplete sentences', even in formal use -- examples: "Yes!" and even "Nor I". In fact, omitting a verb is not unusual in ordinary conversation. If the verb is implied (as when it has just been used), it does not always need to be (re)stated.

The main issue people have is with the use of "ME".

Observation - You are generally supposed to use "I" as the subject (the one who is doing the action), and "me" for the object (the one receiving the action, often after a preposition). - "I gave it to him". "He gave it to ME"

But it has long been acceptable spoken English to use "me" in certain places where strict, formal grammar requires, or supposedly requires "I". The most common example is when answering the question "Who's there?" Some grammarians insist that you must say "It is I". But in most SPOKEN English you will here "It's me" or simply "Me!" And it is NOT considered vulgar or lower class except by certain grammar snobs. (There's also a fairly solid grammatical explanation for why "It's me" works... but I don't want to get too far off track!)

In fact, it seems that "ME" is commonly used for EMPHASIS, even when it's the subject/actor you are emphasizing -- especially if you speak WITHOUT using a verb. Thus, when you are giving a ONE-word answer (to "Who's there"? or "Who made this mess") and NOT using a verb, you simply say "ME". Note that goes for ALL the pronouns "Him" "Her" "Us" all work the same way. "Me neither" (no verb!!) follows the same pattern.

As for "NEITHER" vs. "EITHER" -- generally, I'd encourage you to say "neither".

Suppose your friend says "I didn't like that movie" or "I don't want to go home yet". Your response could be the formal "Nor I"/"Neither do I". But colloquial usage would often be either "I don't either" OR the simple "Me neither".

The point is this. Since you are communicating a NEGATIVE idea, you should include a negative form. "Neither do I", "Nor do I", "Nor I", "I don't* either" and "Me neither" ALL do so. So does another alternative that may help this discussion -- "Not me, either." (This last form, however, is a bit more likely to be used when me IS the object.)

So what about "me either"? Well, you do now find that form in American English. Though I do not recommend it, I would not worry about it too much. I believe it began as an abbreviated form of "Not me, either".

In any case, when someone uses that form, it is CLEAR that they are AGREEING with the negative of the person they are answering, so there is no confusion about the meaning. (That's why, though I don't recommend it, I don't worry about it either.)
I believe it to be "me neither," because "me" ends in a vowel, therefore signifying that the next word should really start with a consonant. Just saying "me either" sounds as if it is defying the rules of the English language. Hope this helps! ;-)
Almost always, neither is correct. Those responses are, typically, comebacks to something like "I don't know him". A proper response would be "Nor I" as a shorthand for " "Nor do I". Would you, actually, say, "Nor do me"? Well, given the atrocious state of grammar in the US today, I suppose you might.
me either is the correct way . me either doesnot make any meaning correctly.
The correct answer is based according to what sentence you're using. Either gives you a choice.
Ex. : I will go to the fair with either my sisters or my friends.

In the sentence above, you have to decide with whom you are going to the fair.

Neither does not give you a choice. It is basically a made decision.
Ex. I will go to the fair with neither my sister nor my friends.

Now, in this sentence above, your decision has already been made for you.
if you're talking about proper English, neither is correct because they aren't proper sentences. if you're talking about possible responses, it would depend on the whether you want or don't want something.

would you like a slice of cake or a cookie?
Answer 1: erm... me? neither, thank you. (proper English response--thank you, but i would not want any.)
Answer 2: erm... for me, either can (proper English response--i wouldn't mind either one, thank you.)

so it depends on whether you don't mind both choices or you don't want both choices :D got it?
Well, to be technical, neither are correct. They are both incomplete sentences. There is a subject, but there is no verb. If you are alligning yourself with a negative statement, your complete sentence might be "I do not agree with that statement, either." So I think the confusion is that we are trying to make something "correct" that is by its very nature, incorrect grammar.

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