Me neither would be used to agree with a negative statement that someone has just said such as 'I don't like chocolate' and then if you also don't like chocolate you could say 'me neither.' Either this or that is used when there is a choice between two different plans of action, but both together are not possible, it's an either-or situation - we can buy a new TV or we can go on holiday, but we can't do both. Either do I is not really used.