Squire graduated B.A. at Balliol College in 1560, and became a Fellow that year, graduating M.A. in 1564. He became vicar of Cumnor in 1568, and accumulated other preferments, being canon of St Paul's Cathedral in 1577.
Squire's suspicions of Robert Persons were instrumental in forcing Persons, who was Dean of college, to resign his Balliol fellowship in 1574. They had clashed when Persons was Senior Bursar in 1572–3. Squire had an ally among the Fellows in Christopher Bagshaw.
Squire himself had a reputation for dealings with the supernatural. It was alleged against him by Persons that he sold familiar spirits, in the form of a fly, to gamblers; or, in the term "dycing flies", the word fly was then a synonym for familiar. The charge came close to losing Squire his post as Master. Richard Harvey, in defending his own practice of astrology, mentioned Squire among other academics as sympathetic to it. According to Balliofergus he was "a great Mathematician".
Squire embezzled a legacy given to the college. Around 1580 he was paid off by the Jesuit George Gilbert to turn a blind eye to the development of a Catholic association of young men in the area (Farringdon Without) of Chancery Lane or Fetter Lane. In 1588, the year of the Spanish Armada, and also his death, he was given custody of a leading recusant, Walter Fowler.
Squire married a daughter of John Aylmer, the bishop of London to whom he was personal chaplain, by 1587. Their son John was a Cambridge graduate, and became vicar of St Leonard's, Shoreditch.
John Strype relates that Squire preached his own wedding sermon, that he was unfaithful to his wife, and that Squire fabricated an affair she was having. Finding out about this, his father-in-law the bishop "cudgelled" him. Further, he ran up debts, and his estate was put into administration. John Squire was brought up by Theophilus Aylmer, son of the bishop.
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- Richard Bauckham, Science and Religion in the Writings of Dr. William Fulke, The British Journal for the History of Science Vol. 8, No. 1 (Mar., 1975), pp. 17-31, at p. 23. Published by: Cambridge University Press on behalf of The British Society for the History of Science. Stable URL: http://www.jstor.org/stable/4025814
- Frances de Paravicini, Early History of Balliol College (1891), p. 324; archive.org.
- archives.balliol.ox.ac.uk Balliol College Archives & Manuscripts.
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- "Squier, John (SKR600J)". A Cambridge Alumni Database. University of Cambridge.
- Ralph Churton, The Life of Alexander Nowell, Dean of St. Paul's (1809) p. 297; archive.org.
- John Strype (1821). Historical Collections of the Life and Acts of the Right Rev. Father in God, John Aylmer: Lord Bishop of London, in the Reign of Queen Elizabeth, Wherein are Explained Many Transactions of the Church of England and what Methods Were Then Taken to Preserve It, with Respect Both to the Papist and Puritan. Clarendon Press. pp. 123–6. Retrieved 16 July 2013.