Squire made his debut at a chamber music concert given by Albéniz at St. James's Hall in London on 12 February 1891. He was associated with the concerts of British chamber music which were given at the Queen's Hall, London in 1894. On 20 April 1895 he appeared at the Crystal Palace in London to play the Saint-Saëns cello concerto in A. He was appointed principal cello at the Royal Italian Opera (now the Royal Opera House Covent Garden) between 1894 and 1899. As a prominent member of the original Queen's Hall Orchestra from 1897 to 1901 he played in some of the earliest Henry Wood Promenade Concerts; there he performed in some of his own compositions, for example "Serenade" in 1897 and pieces by other composers, for example the Andante from the cello concerto in D major Op. 45 No. 2 by Molique in 1898. Henry Wood said, of Squire: