"neither" and "nor" and "don't either" etc are for use in response to a negative. ("I DON'T eat meat")
"either", unless directly preceded by "don't" or some other negative, refers to alternatives, such as "You either eat meat or you don't."
Strictly speaking "I" is used when the speaker is the subject ( you wouldn't say "me don't eat meat")
"me neither", strictly speaking, is short for "me also don't", which has "me" as the subject. However, hardly anyone says "I neither" and "Me neither" has come to be acceptable. "Nor me" is still less acceptable than "Nor I" in this context.
"Me neither" or "Nor me." would be a response to the speaker being the object, such as when the initial statement has the form "Dogs don't bite me."
"I don't bite dogs." could be answered by "Nor I."
"Dogs don't bite me." could be answered by "Nor me."
In response to "Don't bite me!" Be careful not to say "Gnaw me!"