The records for the RC parish of Addergoole start in January 1840. I did search the subscription site Roots Ireland but was unable to find a marriage record for John Callaghan and Bridget McLane. I also could not locate any siblings to Anthony.
If John and Bridget were from Addergoole parish baptismal records will not be available.
The 1855 Griffiths Valuation head of household listing does show two John Callaghan records. Possibly one of the two records was for your ancestor. https://www.failteromhat.com/griffiths/mayo/addergoole.htm
Did the family stary in Ireland?