I think the example is correct. I was in the process of writing a script for a local news broadcast and I was curious which was the proper, "me neither" or "me either". I thought it was the second but I have often heard the first used and also have seen it typed online. But when I said it in my mind I thought something sounded...off. So I looked it up. It may not really be considered a big deal in most every day life, however, I do not wish to use improper english in matters that are meant to be taken seriously and which are professional. So I will be using the second.