She attended Ohio Wesleyan University and, after graduation, began acting in regional theater at the Cleveland Play House.
She later appeared in Broadway productions of Goodbye Again (1932), Hipper's Holiday, Three Men on a Horse (1937), and Lady of Letters (1935), among other plays.
Squire also continued acting on the stage with roles in The Traveling Lady, Six Characters in Search of an Author, The Sin of Pat Muldoon, and The Shadow of a Gunman.
Squire returned to the stage the following year in a co-starring role in the New York production of Roots, by Arnold Wesker.
In 1979, she had a leading role in Hillbilly Women at the Long Wharf Theater in New Haven, Connecticut.
Squire's final onscreen role was in the romantic comedy film When Harry Met Sally... in 1989.