Why do you think they changed their name to Jason from Jennett?Both Peter and Charles are in Franklin Co., NY in 1860, and it seems this is where Charles may have lived for many years after.
1860 census, Bangor, Franklin Co., NY: Peter Jason, age 23, b. VT, occupation day hand Eliza, age 18, b. Canada
Charles Jason, age 28, b. NY, occupation day hand Julia, age 20, b. Canada Lucy, age 9 mos., b. NY
In 1870 Charles was in St. Lawrence Co., NY, but in 1880 back in Franklin Co.
There is a book which came out in November, 2013, titled "Franklin County Soldiers Serve in the Civil War -- A New Book", compiled by Maren E. Dodge Miles.It might and probably does have info on your men.You can Google the title to find out where to purchase it.