He attended the Istituto Musicale, Turin, for six years, studying violin under Francesco Bianchi and Pietro Bertuzzi, and composition under Carlo Pedrotti.

He established a performing and teaching partnership with fellow Italian Faustino Ziliani and featured in many local concerts.

Squarise was associated with the Adelaide String Quartet Club, St. Francis Xavier's Cathedral, and various local music societies.

In 1887 Squarise married Camille Villanis (née Chapuis), the Swiss widow of an Italian vigneron, at the residence of Adolph Marvale in Norwood.

Some of Squarise's piano compositions were published during the 1890s and in 1894 he composed a comic opera, Fabian, to a libretto by Donald Cargill.

In 1904 Squarise formed the Dunedin Philharmonic Society, an amateur orchestra of fifty to seventy players.

It frequently featured Squarise's own compositions, including his Symphony in C Minor, numerous marches, and the patriotic battle fantasia Military Caprice (1914).