We investigate sectorial operators and semigroups acting on non-commutative L^p -spaces. We introduce new square functions in this context and study their connection with H ^(...) functional calculus, extending some famous work by Cowling, Doust, McIntoch and Yagi concerning commutative L^p -spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative L^p -spaces. We mainly focus on non-commutative diffusion semigroups, that is, semigroups ( T_t )(...) of normal selfadjoint operators on a semifinite von Neumann algebra ( M, t ) such that T_t  : L^p ( M ) (...) L^p ( M ) is a contraction for any p (...) 1 and any t (...) 0. We discuss several examples of such semigroups for which we establish bounded H^(...) functional calculus and square function estimates. This includes semigroups generated by certain Hamiltonians or Schur multipliers, q -Ornstein-Uhlenbeck semigroups acting on the q -deformed von Neumann algebras of Bozejko-Speicher, and the noncommutative Poisson semigroup acting on the group von Neumann algebra of a free group.