Hello Nylon,
I too am researching the White's of White Hall. If you search the RootsChat site you'll get the picture!
You mention that John White had four sons, one of whom was James Robert, the father of George Stuart White (born 1835) : I wonder if you have learned what the other sons were called? I am hoping one was named George.
The George White I am seeking details of, died in County Antrim in 1876. He may have been born anytime between (I guess) 1785 and 1812. His son, Charles, however, was born in County Down in 1832.
Regards, Elizabeth