sources below
go to Elizabeth Leavitt's page (not yet online)

.....There is no apparent record of Joshua's birth. In later Hingham records he is called Joshua, Jr., but this is most likely to differentiate him from an older Joshua in that town. He was living in Abington when he married, and it has yet to be determined if he may have been born there. He seems to have been a bit fluid between the two towns. Although both he and Elizabeth Leavitt were residents of Abington when they married, the marriage took place in Hingham, with a publication of their marriage intentions in Abington.

the rest of this needs to be edited after looking again at the Abington town records and the church records

Their children appear inconsistently in both the Hingham and Abington town records of births. This can be explained by the fact that the births were not recorded as they happened. The town clerk gathered the information periodically, and the records are mostly formed in groups. Joshua and Elizabeth were probably asked for records for their children about 1758-1760. All are accounted for in the Abington records except the first of their sons named Caleb, who had died previously, and Barzilla. The latter omission has no obvious explanation given the appearance of the Beal children in a distinct group and that Barzilla lived to adulthood. He does appear in the Hingham records. In addition to the appearance of these births in the Abington town records, the published vital records for that town also include the records of the First Church. From the vital records it is known that Caleb the first was baptized there, and that is the only reference to him. The birth records for the Beal group in the town records end in 1758, but the baptisms for the later children between 1758. The Hingham records are known not to be complete, and this is surely so of Abington as well, but the original Abington records show the Beal children in a group. Caleb Comparing this with the apparent baptisms in the church in Abington, it appears that after their marriage they moved to Hingham, where Joshua was born in 1745. The next child was Caleb.

children of Joshua and Elizabeth (Leavitt) Beal:(4)

Joshua b. 27 November 1745
Caleb bap. 1 May 1748
Molly b. 18 April 1754
Caleb b. 2 March 1756
Huldah b. 3 October 1761
Barzilla b. 12 March 1764
Comfort b. 20 May 1766