I found this
December 16 1859
December 8, in the Presbyterian Church, Newtowncunningham, by the Rev. William Scott,
GEORGE, only son of MR RODGERS, Tullyannon, to SARAH, eldest daughter of MR ROBERT McGIRR, Ballyhaskey, near Newtowncunningham
June 19 1863, by the Rev. William Scott, in the Newtowncunningham Presbyterian Church, MR FRANCIS YOUNG, Knockglass, Malin, to MISS ELIZA H. ROGERS, only daughter of the late MR ROGERS, of Tullyannon, Newtowncunningham
Link is
http://freepages.genealogy.rootsweb.ancestry.com/~donegal/allsaintsmn.htmAlso from goggle searching appears St. Johnston and Newtowncunningham are neighbouring areas
In Griffiths Valuation for TULLYANNAN, ALL SAINTS, DONEGAL there is a John Rogers probably was meant to be Rodgers. Griffiths was taken around 1857 for Donegal.
Tullyannan Griffiths ValuationNote that Griffiths valuation was not a census. But recorded the head of house for those that held/rented land/property. So there are two Rogers/Rodgers households.
First one with head of household as John Rogers. And second family has head of house as Anne Rogers. Possibly a spinster or a widow as she is the head of house.
If does mention in the marriage notes above that George was the only son of Mr.Rodgers. So would be possible Anne is a mother of the George getting married and she was a widow. Leaving it possible then that John Rogers/Rodgers in Griffiths was Anthony's father.
If you click on original page on that link you can see what land he rented/held etc. you can also see the map and reference from the original sheet can find exactly on map where this John Rogers (Rodgers) lived. John Rogers is in plot two and on the map you can see the outline of the house and bulidings. If you need any help viewing the map let me know.
In the 1911 Census of Ireland there is a George Rodgers living on his own in Tullyannan probably the husband of Sarah McGirr also interesting to note there is a Porter family living there in 1911
1911 Tullyannan CensusThere is some deaths for Rodgers Tullyannan on this link
http://mcn.ie/crossroads/genealogy/Deaths_date.htmlAnd also an interesting link for flax growers in 1796 has some Rodgers of interest
http://www.ulsterancestry.com/ua-free_FG_Donegal.htmlThere is a Sarah Rodgers born 12-May-1876 to George Rodgers and Sarah McGirr in Tullyannon found that on familysearch.org website.
I also found Margaret 1-Nov-1871, Catherine Ellen 20-Jan-1874, Elizabeth 2-Jan-1865 and John 16-Aug-1878.
For the couple Francis Young and Elizabeth Rodgers i could not find any children.
But i did find a George Rodgers born 7-Apr-1879 in Tullyannan to a William Young and Elizabeth Rodgers. They also had William 24-Sep-1870 and Mary 16-Jan-1881.
According to 1911 Census George was born around 1836. A good chance he was a first or second cousin of your Anthony.
Hope this helps
Ger