Birth registration didn’t start in Ireland till 1864 so you won’t find a birth certificate for John. Prior to 1864 you need to rely on church baptism records. You haven’t said what denomination the Black family was.
For Derryvullan, there’s a Church of Ireland in Castle Archdale whose baptisms start in 1842. Records in local custody (ie the Rector has the only copy). The Derryvullen South Church of Ireland’s early records were destroyed in the 1922 fire in Dublin. They have nothing earlier than 1877. Derryvullen North has baptisms from 1803. Copy in PRONI in Belfast. Irvinestown Methodist has baptisms from 1829 and marriages from 1863. (Copy in PRONI). Marriages prior to 1863 are likely to have been in the Church of Ireland. Irvinestown Presbyterian has records from 1842. Copy in PRONI. RC has records from 1846. On-line on the nli site.
For Enniskillen, the Church of Ireland in the town has no records before 1861. The earlier records for the 1800s were lost in the lost in the 1922 fire. (There are some records for the 1700s though. Copy in PRONI). Tempo Church of Ireland has baptisms from 1836. Copy in PRONI. Enniskillen Methodists baptisms from 1823, marriages 1864. (Marriages prior to 1864 are likely to have been in the Church of Ireland). Tempo Methodists have baptisms from 1841. No marriage records. RC Enniskillen has baptisms from 1831 and marriages from 1817. Tempo RC has baptisms & marriages from 1845. On line on the nli site.
The records in PRONI are not on-line and so you need to go in person to view them. (If you can’t do that you may need to employ a researcher).
Regarding the conflict between the place of birth, death certificates often have mistakes because the person recording the information won’t always have first hand information. Birth certificate and marriage information tends to be more reliable, so I’d favour Derryvullan over Enniskillen. (Enniskillen is about 20 miles from Irvinestown it the Derryvullan area and is the biggest town in the county, so the two are not incompatible, if describing where you came from to someone in another country.
I looked in the 1901 census for Co. Fermanagh. There were 31 people named Black. 1 Presbyterian and 1 Methodist. The rest were Church of Ireland. (So all the Black families in the area were almost certainly descended from settlers from Scotland or England who moved to Ireland in the 1600s, as part of the Plantation).
Though most Irish censuses pre 1901 have been destroyed, a few fragments exist. That includes part of Co Fermanagh. The 1821 census of Derryvullan has survived. 2 Black families shown:
http://www.census.nationalarchives.ie/pages/1821/Fermanagh/Derryvullan/Irvinestown/73/
http://www.census.nationalarchives.ie/pages/1821/Fermanagh/Derryvullan/Ballindullagh/14/