Here's a town history talking about 1812.
Goshen Town History

This one's interesting. It talks about 1812, as well and then mentions a legend that a frame built in 1801 in Antrim, NH was square ruled.
Newbury, VT town history

Here's one that gives 1815 as the date of the first local square rule frame -
Antrim town history

Here's one written in 1875 that claims 1760's as the first square rule frame in town. That seems too early to me.
town history of Northfield, Mass

Here's one that mentions the first carpenter in town to use the square rule as being a veteran of the War of 1812.
Washington Co., Vt.

Here is an interesting passage from a James Fennimore Cooper novel from 1845.
The Chain-bearer

Nothing conclusive about the true origins, but interesting stuff about when it started showing up in different places.

Gabel