MPassaro -- CONGRATS on your Enameled Square.
This happens to be one of my favorite early Hamiltons, simple but cool. Though, I am a bit of a Square!
At last count I own 10 different Squares, including: GGF Enameled Square, 14k WG Enameled Square, 14K plain Square early model, 14K white Square early model, 14K 2-tone plain Square later model, wgf-engraved Square, ggf-e Square B, 14K WG Square B plain, 14K green engraved Square, and a recently located and somewhat of a mystery early plain green (or perhaps yellow?) Square that is marked gold filled but has a matching # center piece marked "base metal". Among these are variations without a second hand, with sub-6 second, and with a sub-9 second. Please see an older photo that included my 2 enameled Squared on the left -- the white one now has a much nicer dial.
I think that some of the confusion around this watch is related to how they were shown in Rene's terrific early book "Hamilton Wristwatches: A Collector's Guide". This model is the only one shown on the cover. My guess is that when the book was written, he carefully made some of the best possible guesses he could make, but information on some of the earliest models (this Square, the O-sized wristwatch, and the Rectangular too, among others) was simply not yet known. In this book, the Enamel Square is correctly illustrated as a solid gold and also a gold filled model. However, it was incorrecly labeled as the "Square Engraved". Because of this, the manufacturing numbers were also presumably overstated, especially those listed as yellow, white and green gold-fill -- I PRESUME that these numbers were actually the much more common Square Engraved models. (Keep in mind that the average Oxford Dictionary has a few dozen errors also!)
I would like to see the mystery and manufactured quantities of the early Square and the later Square (with the larger center) resolved a bit more also. I PRESUME that the early Squares did not come with engraved bezels, but only the later ones did.