Hello.
As you already know, shoemaker George June was in the 1850 US census for Carlton, Orleans County, NY.

It appears that there may possibly have been another brother, Amos L. June, a shoemaker who was born in CT about 1825-27.

In the 1860 and 1870 US census for Carlton, Orleans County, NY Amos is there with his wife Deborah and son Fred L. and in the 1880 US census he is a "shoe merchant" still in Carlton but with a second wife named Jane and a 3y old daughter Geneveve.

Amos L. June died in Waterport, Orlean County, NY in 1894 (see below). If all else fails, it might be worthwhile to send for the death certificate of Amos to see if HIS parents are named in it and then see if you can somehow link them to George and Benjamin June.

Rochester Democrat and Chronicle
Monday, January 15, 1894; page 4
Orleans
The death of Amos L. June, of Waterport, occured Friday after a protracted illness with paralysis. The deceased was an old resident of this town and a well-known business man. He was sixty-nine years of age and leaves a son, Dr. F. L. June, a daughter, Jennie, and a wife.