Born in Florence, Enriquez was the son of conductor Vittorio Gui and musician Elda Solaroli.

[1] In the late 1940s, while being a student of Italian literature at the University of Florence, he became assistant director of Giorgio Strehler and later of Luchino Visconti.

[1] In 1951, he made his directorial debut with an adaptation of the George Bernard Shaw's drama Caesar and Cleopatra, played by the Ricci-Magni stage company at the Teatro Eliseo in Rome.

[1] Starting from 1954, Enriquez was also very active on television, specializing in adaptations of stage works, notably Shakespeare's Romeo and Juliet and Antony and Cleopatra.

[1] He later became artistic director of the Teatro Stabile di Torino and then of the Teatro Stabile di Roma, and in 1968 he co-wrote with Franco Cuomo and directed one of the first Italian works inspired by the Protests of 1968, the drama play Discorso per la lettera a una professoressa della scuola di Barbiana e la rivolta degli studenti ("Speech for the letter to a teacher from the Barbiana school and the students' revolt").