Hi I have worked on my husbands Enright's and I am pretty sure they are from County Kerry, but the interesting thing is the ships passenger list of the Ship Carack out of Liverpool, landed at the Port of New Orleans, November 29, 1851.
Search on ancestry under Emright if you can't find it under Enright. My husband's ancestor Robert Enright is on the ship with his parent's Matthew and Nancy, also Bartholomew 18 and Ellen Enright 20. The interesting thing is a John Kelly (15 years old) is traveling with them. Occupation weavers from Ireland. They ended up in Hancock County, Indiana. Three other grown sons of of Matthew an his wife Nancy (also known as Ann) had come a few years earlier to work on the railroad in Dearborn County, Indiana. I don't know if they are any relation to your Enright's but found some free online Parish records from county Kerry at Baptisms of Causeway Parish, Diocese of Kerry, Ireland; http://www.irishgenealogy.ie/index.html