Squire was commissioned lieutenant on 20 September 1765, seeing service off North America in the Mermaid 28, Captain James Smith. He was promoted commander of the Bonetta 10 on 21 January 1771 whilst still serving in North America, and this vessel was paid off in August 1772.
He recommissioned the sloop Otter 14 in January 1775, going out to North America in March and serving off the coast of Virginia later that year where, on 24-25 October, he led an aborted attack up the James River on Norfolk with a number of small craft manned by his own crew and local loyalists. After being sent to Antigua for a refit the Otter returned to Vice-Admiral Lord Howe?s North American fleet in the summer of 1777 and was present in the Philadelphia Campaign from August-November. Squire retained the Otter until the spring of 1778, and by the summer was commanding the Fortune 16 in home waters.
He was posted captain on 6 November 1779 and recommissioned the Ariadne 20 for service in the Downs. On 30 April 1780 he was the senior officer of a squadron that gave chase to three French privateers of 20 to 24 guns off Flamborough Head. Regrettably only one of his consorts, the armed ship Queen 20, Commander Richard Trotten, joined him in the pursuit, with the result that the privateers were able to escape by taking to their sweeps, leaving the Queen much damaged. The two commanders who failed to join battle, Commander Alexander Agnew of the Fury 16, and the elderly Commander Stephen Rains of the armed ship Loudoun 20, were cashiered following a court-martial, whilst Squire was honourably acquitted of failing to do his utmost to bring the French to action. On 14 June the Ariadne took the privateer Princesse de Robecq.
In July 1781 Squire commissioned the new Astraea 32, going out to North America once more in October and assisting the Diomede 44, Captain Thomas Lenox Frederick, and Quebec 32, Captain Christopher Mason, in the capture of the American frigate South Carolina 40 off the Delaware on 20 December 1782 following an eighteen-hour pursuit. He later took the privateer Julius Caesar in company with the Vestal 28, Captain William Fox, and Duc de Chartres 18, Captain John Child Purvis, on 15 March 1783 prior to being paid off at the beginning of 1784.
He remained without employment until he joined the recommissioned Solebay 32 in November 1789, going out to the Leeward Islands in March 1790 and being paid off in 1791.
After a further period of unemployment he joined the recommissioned Magnificent 74 in the summer of 1795 for service in the North Sea and Channel, being paid off in March 1796. He was appointed to the Atlas 98 in March 1797 and was aboard her when the mutiny broke out at Plymouth on 26 April 1797, his men immediately joining the insurrection. He continued to command her until January 1799 and his imminent elevation to flag rank.
He was promoted rear-admiral on 14 February 1799, and died on 22 January 1800 at Durham Place, Chelsea, being buried at St. Luke?s Church, Chelsea.
Admiral Squire was a friend and early patron of Rear-Admiral Alexander Fraser.