You don't give enough specifics for me to be sure that what I think is the explanation actually is correct, but this is what I think is going on.
The Mayflower Descendant was a periodical which published information which went far beyond Mayflower lineages.Many public and private records were transcribed and published, and these included the vital records of the town of Middleborough.
So my guess, unless you know that the periodical actually stipulates that the ancestor is "of Mayflower descent,"is that your ancestor is included in the Middleborough Vital Records because he resided in or perhaps married someone who resided in Middleborough.
This in no way precludes a Mayflower descent.There are an estimated 35 million or more living Mayflower descendants, and the likelihood that someone who lived in Middleborough during the years covered by the VR would have a Mayflower connection is fairly high--which I suppose is why the records are in the periodical.
The only way I know of to find out is to keep working back, generation by generation.But if I were doing that I'd certainly try all the shortcuts I could think of, with the understanding that any lineage I found would have to be verified carefully.I'd check all the easy online resources such as Rootsweb's WorldConnect and familysearch.orgThere are many more.There's a chance someone has already done the work, and then you can check to see if it was done right.Mayflower ancestry or not, it's bound to be interesting and worth knowing, and can be a lot of fun.