Hello Bernie,
There is indeed a graveyard at Eskra (Eskragh) RC Church, I presume it is quite old as the 'History From Headstones' project has collected data there. It is in the townland of Lisnarable.
My own 'Guide to Church Records', published by PRONI, makes no mention of any registers for Eskra. However, I know that the church was originally part of Clogher and suspect that most older birth/marriage records will be in the Clogher registers.
For example, in Aughnacloy the quite large 'flock' there was under the pastoral care of a Curate-in-charge. He could baptise, marry etc. but transcribed all such events into the register of his immediate superior, the Parish Priest at Caledon. Eskra is now a parish in its own right and probably has a parish register now, but it not be old enough to have any research value.
Clogher RC registers are held at PRONI and contain baptisms 1856-81 and marriages 1825-35 & 1840-81.
Brian