Pat, I believe I know the lady that you mention. I have been helping her on her family. The Thomas Burns she is descended from is a son of Edward who is the son of Elijah Burns. Her Edward Burns may have had a brother named William and he may be the one in Williamson/Franklin County Illinois. I 'll have to do some more digging.
I have done some looking for your William Burns. I found him on the 1870 census of Lasalle County Illinois in Otler Creek Township. I also found the Family in the 1880 census of Morgan Township of Mercer County Missouri he said he was born in Virginia and father born in Scotland and mother in Pennsylvania. With this info I would say that your William isn't related to the Burns of Williamson and Franklin cOUNTY Illinois which is Elijah and his sons.
I haven't found the family in 1860 or 1850 yet to determine who your William parent are.