  We establish local H\"older estimates for viscosity solutions of fully
nonlinear second order equations with quadratic growth in the gradient and
unbounded right-hand side in $L^q$ spaces, for an integrability threshold $q$
guaranteeing the validity of the maximum principle. This is done through a
nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly
elliptic Isaacs operator and perturbed by a Hamiltonian term with natural
growth in the gradient. As a byproduct, we derive a new Liouville property for
entire $L^p$ viscosity solutions of fully nonlinear equations as well as a
nonlinear Calder\'on-Zygmund estimate for strong solutions of such equations.
