All from familysearch.org. https://familysearch.org/ where there is more census data and birth records. The James Quigley b. 1907 in Woburn, MA lists parents as James Quigley b. Brooklyn, residence Brooklyn, bricklayer,and Mary Mullen b. Brooklyn. In 1870 James and Margaret are living in Chelmsford, MA with seemingly the correct children by names/dates. There is a James Quigley, living in Chelmsford, b. Dec. 25, 1840 in IRE who bacame naturalized in 1878 - can't tell if this is the same James or just conicidence? as in 1875 James and Margaret appear to be living in NY.
You might want to try entering the childrens names in the free search tool at the URL above to see if you can figure out more.
James Quigleymale 44 Elijah St,, Woburn, MA Birth06 Oct 1907Woburn, Massachusetts registration date: 1907 registration place:Woburn, MA (p 770 no 264) recorded date: Dec 17, 1907 father: James Quigley Bricklayer, ResidenceBrooklyn, NY birthplace: Brooklyn, NY mother: Mary Mullen birthplace: Brooklyn, NY
New York, State Census, 1875E.D. 2, Ward 7, Brooklyn, Kings, NY
selfJames Quigley M 34 IRE abt 1841 wifeMargaret Quigley F 34 IRE dauMargaret Quigley F 10 MA dauMary A Quigley F 8 MA sonJohn Quigley M 6 MA dauElizabeth Quigley F 4 MA dauSarah Quigley F 2 MA sonJames QuigleyM 0Knights
United States Census, 1870Chelmsford, Middlesex, MA
James Quigley M 27y Ireland Margaret Quigley F 27y Ireland Margaret QuigleyF 5y Massachusetts Mary A Quigley F 3y Massachusetts John Quigley M 1y Massachusetts
United States, New England Naturalization Index, 1791-1906