She had a lifelong relationship with artist Ethel Mars, with whom she traveled and lived in the United States and France.

[3] Squire studied at the Art Academy of Cincinnati from 1894 to 1898;[4] her instructors were Lewis Henry Meakin and Frank Duveneck.

[4] A joint exhibit of works by Squire and Mars was held at the Mary Ryan Gallery in New York in 2000.

[4] Squire and Mars were great friends of Gertrude Stein and Alice B. Toklas while living in France, and the writer's poem "Miss Furr and Miss Skeene", believed to be the first such work to use the word "gay" to describe homosexuality,[10] is meant to describe the couple.

Squire died of heart failure there on October 25, 1954,[1] and is buried with Mars, who survived her,[10][12] in the town cemetery of Saint-Paul-de-Vence.