According to History of Lancaster County, PA:
"In 1732 Ulrich Shirk with 6 sons and 2 daughters came from Switzerland… [from Sumiswald, Switz by way of a few years in Holland] They went to Lancaster for a year. [They actually arrived on the ship Mortenhouse in 1728]
In 1733 three of the sons, Ulrich (Jr), Casper, and John, with their father, Ulrich, settled in the vicinity of Schoeneck, taking about 1500 acres. The other 3 sons, Michael, Joseph and Peter, one settled near New Holland, one along the Conestoga and the other near Mount Joy.
Ulrich [Jr] Shirk reared a family of 3 sons and 7 daughters:
Ulrich [III], born May 16, 1751, died May 14, 1842, married Maria Eberly
David, married Barbara Erb
Michael, born May 26, 1758, died Apr 4, 1827, married Barbara Flickinger
In 1780 we find that Ulrich [III], David, and Michael, sons of the orginal Ulrich [Jr] Shirk, were the only Shirks then living in the township of Cocalico, which proves that [their uncles]Casper and John had already sold their property and left for other parts...."
This source ties Joseph Shirk of Mount Joy, Hempfield twp, Lancaster Co, PA to Ulrich Schürch of the 1728 ship Mortenhouse.
It may be possible to confirm family relationships with Joseph by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share DNA with Joseph: