There is no Snowbird Clan in the Eastern Cherokees, but there is a small portion of the reservation of the Eastern Band of the Cherokee known locally as the "Snowbird reservation". It is a tract of land in Graham County, NC (about an hour or so from present day Cherokee, NC.) These Cherokee are known locally as "the Snowbird Cherokees" but that is simply a local term and not an official designation to my knowledge. The large reservation is in Swain and Jackson Counties in NC. It is the Qualla Boundary Reservation of the Eastern Band of the Cherokee. Present day Cherokee, NC, lies on this reservation. There is no Qualla County in NC. (The town of Cherokee, NC is not to be confused with Cherokee County, NC, which lies about 1.5 hours to the southwest of the reservation.) If your folks were in Henderson County in 1880, they were at least 2.5 hours away from Snowbird by today's driving times. If you will e-mail me with the names, I will check the census for them in 1870 in all the NC counties.
There were Featherstones in Buncombe County, NC (all the western counties were formed from Buncombe, NC and/or bordering Indian Lands) between 1792 and 1800. They are in the land records and Meriman Fetherston was on the 1800 Census. That would indeed make them among the first settlers of western NC, but means that these Fetherstons were not Cherokee (although one or more Fetherston/Featherstone may have taken a Cherokee bride.)
Hope this helps.