Squire was the son of a druggist in Warminster, Wiltshire, and was first educated at Lord Weymouth's Grammar School.
He matriculated at St John's College, Cambridge in 1730 and graduated BA in 1734, winning the Craven scholarship the same year.
Adding to his growing number of parish livings, he was appointed rector of Toppesfield, Essex (1749–50) and subsequently of St Anne's Church, Soho (1750–66), and vicar of St Alphege's, Greenwich (1751–66), where William Paley, who later achieved fame as a theologian and philosopher, served as his curate.
His Letter to a Young Gentleman of Distinction (1740) argued for the benefits of a standing army against a militia to protect Britain in its wars against France and Spain.
[4] In May 1746 Squire was elected a Fellow of the Royal Society as: "A Gentleman well known to the Learned World by Several valuable Treatises, perticularly [sic] 'Two Essays on the Antient Greek Chronology' and 'On the Origin of the Greek Language'; A new Edition of Plutarch's Discourse on 'Isis & Osiris', with an English Translation & Commentary; and an 'Historical Essay on the Anglo-Saxon Government both in Germany & England'".