Fiona,
Searched for birth/marriage information on William and his parents. In Ireland, the Indexes to the Civil Records of Irish Births, Deaths, and Marriages date from 1864 with non-Roman Catholic Marriages recorded from 1845. The transcription website
www.rootsireland.ie has a large holding of early church records but a search has not been productive. Do you know the religious affiliation?
The surname Green/e is more common in other parts of Donegal, When searched 1858 Griffiths' valuation found that in Fahan which is close to Buncrana, there was a Daniel Green, and on the nearby island of Inch a William Greene. Fahan also has a William Little recorded.
https://www.askaboutireland.ie/griffith-valuation/
There are Inishowen groups for DNA matching (on Facebook) which could be of assistance in finding connections if have taken a test.
New Zealand is not such a common destination for early emigrants, is there any family history of traveling with others from the locality?
I will check further for information here.
Regards,
Marion Whelan - IrelandXO Volunteer ☘
I
.
.