We consider f(R) = R + R^2 gravity interacting with a dilaton and a special non-standard form of nonlinear electrodynamics containing a square-root of ordinary Maxwell Lagrangian. In flat spacetime the latter arises due to a spontaneous breakdown of scale symmetry and produces an effective charge-confining potential. In the R + R^2 gravity case, upon deriving the explicit form of the equivalent local "Einstein frame" Lagrangian action, we find several physically relevant features due to the combined effect of the gauge field and gravity nonlinearities such as: appearance of dynamical effective gauge couplings and confinement-deconfinement transition effect as functions of the dilaton vacuum expectation value; new mechanism for dynamical generation of cosmological constant; deriving non-standard black hole solutions carrying additional constant vacuum radial electric field and with non-asymptotically flat "hedge-hog"-type spacetime asymptotics.