Westerns are their own genre, known as such, and there are hundreds to thousands of examples.
Many movies have been made with Southern themes and set in Southern locales, replete with Southern accents, no shoes, and the lack of any redeeming social value.
I guess the Canadian and Alaskan themed things could be seen as Northerns.
But what are some of the better Easterns? (USA, not Europe or Asia)
I think a few minor filmmakers have made films set in New York City.
Hair, for example.
Quite true. A large percentage of those would be “present day” (whenever they were made) and somewhat lacking in any of the period texture that many Westerns have had. Maybe if we exclude the crime dramas of present day settings, we could get more of the Eastern flavor.
Either that or move away from NYC to some degree. Maybe avoid the big cities, too, like Boston, Philadelphia, Washington DC, Baltimore, and such.
By comparison, very few Westerns take place in the bigger cities, unless you go to counting Tombstone, Dodge City and the like.
Manhattan
Annie Hall
Radio Days
. . . pretty much most of the Woodster’s films. 
Last of the Mohicans
Witness
On Golden Pond
Sleepy Hollow
Rocky
Groundhog Day
Dead Poets Society
Shag
Runaway Bride
My Fellow Americans
I thought Bedford Falls from “Its a Wonderful Life” is set in upstate New York (well, upstate of NYC anyway).
For nineteenth-century New York City, how about:
The Age of Innocence
Gangs of New York
The War of the Worlds
All of the John Waters movies set in Baltimore.
Drums Along The Mohawk
Captains Courageous, which is getting a bit far east
Conrack: The Water is Wide
A Few Good Men
The American President
Thank You for Smoking
Pocahontas
The Patriot
Martin Guerre: The Return of the Soldier
Sommersby
Just something to consider when ascribing the Eastern designation:
Is the Mississippi River a fair demarcation point for Eastern vs. Western?
Are the Confederate states the Southern area?
Just asking. No response needed. Just make sure Eastern is the dominant geographical consideration.
Quoted for emphasis. “Western” generally applies to a film set in the American West during the period of westward expansion. There are exceptions, but they usually incorporate tropes common to the historical western. Not every film set West of the Mississippi is a “western.” By analogy, an “eastern” should be a film with set in a location east of the Mississippi during a period of historic transformation.
That seems to me an unnecessary restriction. Westerns don’t avoid big cities, they take place in a setting in which city life was not the norm (although perfectly good Westerns have used a bustling San Francisco as a setting). The history of the East, however, is largely the story of the growth of cities such as New York, Boston, and Philadelphia.
Gone with the Wind
The Great Eastern, of course.
Once Upon a Time in America
Also, if it’s based in a Stephen King book then it happens in New England, possibly Maine.
Nobody’s Fool
Empire Falls
The World According to Garp
I came in to say Last of the Mohicans, but was beaten to it. Let me elaborate and say that it is a great “Eastern Western”. It takes place in the days of westward expansion, but when the Appalachian Mountains were the western frontier. It has all the elements: the beleaguered and increasingly desperate natives whose lifestyle is already extinct, but they don’t know it yet. The frontiersman. The Ladies in Distress. The unruly, and yet sympathetic, villain (Wes Studi knocked Mogwa out of the park). Gunfights, brawls, scenery. Everything.