The John Neary you have found on the 1905 NY State Census is the brother, or possibly the father of Elizabeth McGurk. His wife and children are the same as on later Census returns (minus Anna Rita who was born after 1905). John is the son of Thomas b.Ireland, d.1882 and he himself had a son Thomas b.Chicago in 1895.
Interestingly, as someone else as pointed out, the birth record for Thomas Neary gives his parents names as John J Neary and Anne McGuire - could she possibly have been Ann McGurk?
Someone has kindly offered to look up the Baptismal Record of William Neary on Tuesday and hopefully that may give some more information