Based on father's name from the census, there's a possible match on the extracted births for on of those index records I mentioned :
The BMD Index reads :
Name: Michael Eagan
Registration district: Athy
Event type: Birth
Year: 1877
Volume: 3 / Page: 443
see :
https://familysearch.org/pal:/MM9.1.1/FBS1-BZPand the extracted record :
Name: Michael Eagan
Birth date: 03 Feb 1877
birthplace : Athy, Queens Co. (district)
Parents: Michael Eagan & Julia Stapleton
Ref.: v 3-2 p 443
see :
https://familysearch.org/pal:/MM9.1.1/FG6K-TBP(area/district and page number correspond with the index , so a match)
The extracted record mentions a 'Father's Birthplace' as Inch - this is incorrect, as this detail is not included in birth registers or certs, but may relate to place of birth, or father's residence.
A birth cert would narrow down the place of birth - likely the name of the townland.
Shane