I stumbled upon what looks to be a very well sourced article from NYGBR 146:1 (2015) on John Ogden of Rye. In it the author says that John married ca 1661, Judith Budd who was born about 1640. The author says that Judith's parents were John and Katherine (Brown) Budd of Southold and Rye. He cites John Budd's will, dated Oct 15, 1669 at Rye, which left legacies to his three children including John Ogden and Judith his wife. The author cites numerous other sources in his article as well, but on initial review, I do not see anything further regarding Katherine Brown's origins.
The author refutes some of Baird's work on the Ogden family, and says that John was "possibly" a brother of Richard Ogden of Stamford, Hempstead and Fairfield, likely born about 1640, place unknown, and died in Rye, New York before 1683 when Judith remarried. He refutes that he is a son of Richard Ogden of Fairfield as claimed by Baird.