We present new parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations on both simplicial and rectangular meshes using a velocity gradient-velocity-pressure formulation. We obtain an optimal L2-error estimate for the velocity in both the Stokes-dominated and Darcy-dominated regimes. Moreover, thanks to H(div)-conformity of the velocity, the velocity error estimates are independent of the pressure regularity.