This is a multi level problem, so worth addressing in some detail.
First, in the 18th century "Sr." and "Jr." are often used to distinguish men of the same name living in the same area, even when they are unrelated.
Secondly, in the 18th and 19th centuries, fathers and sons are occasionally distinguished using "Sr." and "Jr." even when the father (apparently) had no midddle name but the son did. Most experts consider this to be a transitional period, where usage was driven by the primary given name, and this usage continues colloquially even into the early 20th century.
Thirdly, there seems to be no contemporary evidence Squire Jr. had the middle name Maugridge, despite wide usage across the Internet. We would, I think, want to find primary sources--contemporary documents--to support his name.
I looked, just casually, for such evidence, and didn't find anything. Maybe someone else will find it, and that would be very cool.
On Findagrave, there are two profiles for Squire Jr. One is his (modern) gravestone. The other is a cenotaph. Neither has the middle name Maurgridge:
https://www.findagrave.com/memorial/18853/squire-maugridge-boone
https://www.findagrave.com/memorial/211422366/squire-boone
There is also a marker for the "Squire Boone Burial Cave." No middle name.
https://www.hhhistory.com/2020/11/daniel-boones-baby-brother-squire...
Then, I looked at some other genealogy websites to see how they are handling this problem.
At FamilySearch, Squire Boone Jr. appears without a middle name. The changelog fshows the middle name Maudridge was removed on Sept. 15, 2023: "Removed Maudridge as a middle name. There is no evidence to suggest a middle name." It was also previously removed on May 13, 2023, January 12, 2023, August 29, 2022, and June 29, 2022,
https://www.familysearch.org/tree/person/changelog/L1SM-5D1
At Wikitree, Squire Boone, Jr. appears without a middle name. This site shows his signature, without a middle name or middle initial:
https://www.wikitree.com/wiki/Boone-116
Similarly, at WeRelate, Squire Boone, Jr. also appears without a middle name:
https://www.werelate.org/wiki/Person:Squire_Boone_%2818%29
Finally, I looked at some of the classic Boone genealogies for evidence of the middle name Maugridge. Nothing.
Boone Society: https://boonesociety.org/squire-boone
Filson Club Quarterly: https://filsonhistorical.org/wp-content/uploads/publicationpdfs/16-...
Spaker, The Boone Family: https://archive.org/details/boonefamilygenea1922spra/page/72/mode/2...
This should be enough to make the point we need to be careful about interpreting names in historic periods, and always double check what we think we know.