Sorry Zacchary, my mistake there are actually 2 William Laybourne in this family, half-brothers, and it is the elder brother who died without issue. However the younger one, all the sources state is his name, nothing else, so I still think that there is no proof other than the same name, that he is the father of Joseph Laybourne.
The other issue is that there appears to be no such place as Black Hadley or Black Hadleigh in County Durham. There is a village named Hedley Hill in Durham and I also know there is a village named Black Heddon in Northumberland, which is the county adjacent to Durham. Trying to find out where Joseph was actually born might be a good next step. Finding quaker records from England might also give more information.