Siobhan,
The RC parish records for Cleenish (the parish which includes Drumbargy & Cornagee) only have baptism records for 1835-1839 and 1859 onwards. Marriages don’t start till 1866. Consequently I don’t think you are going to find Hugh & Bridget’s baptisms, nor their marriage, nor probably the baptism of Denis c 1854. You might want to search what records there are in case you come across any siblings. You will face a similar problem searching for early Green records too. RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
Farmers were usually listed in the tithe applotment records. Those for Cleenish relate to 1834. There were no Gilroys in Drumbargy then so it looks as though Hugh acquired his farm there some time after 1834. There was a Patrick Green farming in Cornagee then, but I don’t know what relation he was to Thomas Green.
http://www.irishgenealogyhub.com/fermanagh/tithe-applotment-books/parish-of-cleenish.php
The 1901 census of Cornagee seems to be missing so I can’t tell you who was farming there then. This is the 1911 census:
http://www.census.nationalarchives.ie/pages/1911/Fermanagh/Gortahurk/Cornagee/517842/
Hugh Greene married Margaret Monaghan in 1903. Hugh’s father was another Hugh.
Griffiths Valuation for 1862 lists 2 Greene properties in Cornagee. One held by Hugh & the other by Thomas.
Hugh senior died 18.12.1896 aged 82. His son Hugh was the informant.
Probate abstracts:
Greene Hugh of Cornagee Letterbreen county Fermanagh farmer died 3 November 1929 Administration Londonderry 6 February to Margaret Greene the widow. Effects £13.
Greene Margaret of Cornagee Letterbreen county Fermanagh widow died 24 November 1939 Probate Belfast 12 January to Bernard McCauley farmer. Effects £97 15s. 5d.
Thomas Green appears to have died in 1876, aged 84. He was replaced in Griffiths by Mary Greene. She in turn was replaced by Frances Greene in 1886. Mary Greene died 11.3.1884 aged 84. Her death certificate describes her as the widow of Thomas Green, farmer. The informant was Frances Green, grandson, of Cornagee.
Frances’ name remains in the Valuation records till 1907 when he is replaced by John McManus. Francis had died by the 1911 census. (I can’t find a death certificate though). He married Mary McSherry on 27.8.1891 and the marriage certificate states his father was Thomas Green. So that appears to make him Mary Alice’s brother.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1891/10676/5890739.pdf
I don’t see a birth in Ireland for Mary Alice Green, so I’d be inclined to accept she was born in England (as the 1911 census indicates). The Thomas who died in 1876 aged 84 is most unlikely to be her father, so I’d suspect he had a son also named Thomas who was Mary Alice’s father. And that that Thomas went to England to work, so accounting for her birth there. Possibly Francis was also born there. (I don’t see a birth in Ireland for him). You would need to search English birth records for Mary Alice and Francis.
Children to Francis Green & Mary McSherry:
Edward 5.8.1892
John 15.6.1894
Mary 12.12.1896
Thomas 1.10.1899 Died Cornagee 12.5.1900.
James 4.11.1901
Widowed Mary and 4 children farming in Woaghternerry in 1911:
http://www.census.nationalarchives.ie/pages/1911/Fermanagh/Enniskillen_Rural/Woaghternerry/515840/
Probate abstracts:
Greene John of Woaghternerry county Fermanagh farmer died 25 July 1932 Administration W/A (d.b.n.) Belfast 23 April to Mary Greene widow. Unadministered Effects Nil. Former Grant Belfast 31 March 1936.
Greene Mary Alice of Woaghternerry county Fermanagh spinster died 28 July 1937 at The Mental Hospital Omagh Administration Belfast 29 April to Mary Greene widow. Effects £18.
Greene Mary of Woaghternerry county Fermanagh widow died 10 July 1942 Probate Belfast 18 August to Francis McSherry farmer. Effects £234 7s.