During the Civil War, Squire enlisted in Company F, Nineteenth Regiment, New York Volunteer Infantry, in 1861; he was promoted to the rank of first lieutenant, and was mustered out the same year.

From 1865 to 1879, Squire was employed with the Remington Arms Company as secretary, treasurer, and manager and purchased large holdings in the Territory of Washington in 1876.

As governor, Squire confronted the difficult challenge of maintaining law and order during the anti-Chinese riots in Seattle and Tacoma.

"[5] Squire withdrew martial law on February 22, 1886, but by then most of the Chinese residents already had been expelled from the Territory, put on a ship, and sent to San Francisco.

Subsequently, at the request of the U.S. State Department, Governor Squire investigated the losses of property by the Chinese residents of Tacoma, Seattle, and the surrounding area.