Welcome to Ireland Reaching Out!
I think the Meenan family lived in Tremoge townland in Termonmagurk civil parish. Carrickmore was a village/town near Tremoge. Immigrants tended to say they were from the nearest town rather than the particular townland where they lived.
Here is the 1901 census record for Michael and two sons Michael and James.
From what I can see on Roots Ireland, Michael and Margaret had a number of children. Unfortunately, the RC parish records for Termonmagurk parish are only available online through 1857. I did find one record for a Patrick Meenan. See transcription below. Owen and Francis likely were born between 1857 and 1863. It is possible that the post-1857 records are available locally.
Name:Patrick MeenanDate of Birth:
Date of Baptism:24-Nov-1856Address:TremogeParish/District:TERMONMAGUIRKGender:UnknownCountyTYRONE
Father:Michael MeenanMother:Margaret Morris