No. So do I, Me too Neither do I, Me neither, I don't either You will hear natives say 'me either' very often but it is not technically correct.

in addition to Claire's idea, you can also use these phrases: "I do too." "so am I." although "either" is itself positive, it's only used in negative statements such as: "I don't either" (= "neither do I"). :)

I think neither are correct. "Either" is inclusive (=one or the other), but "either do I" and "me either" don't make sense from the word order. It should be "neither do I" and "me neither". But yes, you may see those "either" phrases written, proving that native speakers can be sloppy at their own language.

'NEITHER do I' and 'me either' are correct! ‘I don’t like it.’ ‘Me either.’ (= Neither do I). Positive statements: 'I like it.' 'I do, too.' OR 'I like it'. 'So do I.' OR 'I like it.' 'Me too.'