My line of McBurneys go back to the Portaferry/Newtonards area of County Down.
My line came ot this country in 1855 but in their family John McBurney (1715-1779) married Ann Laughlin in 1737. Their grandson Wm. was born there as well and emigrated in 1855 w. my gg/g'father Andrew.
Your William may have been desended from this pair as the family names do seem to reappear frequently.
If you use Ancestry.com they have just added wonderful new databases of marriages, deaths etc for all of the UK and I was able to find all of my ancestors there including the grave of Ann and John.