G'day,
obviously pictures don´t allow for more than a rough guess, but to me it seems there´s no connection with the brand of Jaquet Droz at all.
The pictured watch seems to be a mass-produced piece dating around 1910´s to 1920´s, which rules out this one to be made by Pierre Jaquet-Droz (who died in 1790).
The case doesn´t really fit and the mentioned inscriptions point at a case of US production, which was usual practice then; swiss movements and US casework kept customs low and thus are not rarely seen. The engravings on the movement are english language, which indicates a movement for export purposes.
The execution itself is about average, as far as the pictures suggest; probably not the cheapest thing (which then would have less or no jewels), but not a high-end or Haute Horlogerie watch as well.
As per the name, "Droz" is not uncommon in Switzerland.
Given there have been Droz families around for centuries, it is not unlikely someone of this name put his label on the piece; but that doesn´t indicate a particular make or special merits, as it was not uncommon to use such "trade labels" for a long time.
Greetings from Germany,
Peter