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The essentially non-perturbative vacuum polarization effects, caused by an extended external supercritical Coulomb source, are explored for a planar Dirac-Coulomb (DC) system with strong coupling (similar to graphene and graphene-based heterostructures). Taking account of results, obtained in [1] for the induced charge density ρV P ( r), in the present paper the evaluation of the Casimir (vacuum) energy EV P is presented. The main result is that for a wide range of the system parameters in the overcritical region EV P turns out to be a rapidly decreasing negative function ∼ −Z 3 /R0 with Z , R0 being the charge and the size of the external source. By an explicit calculation the possibility for complete screening of the electrostatic reflection self-energy of the external source by such polarization effects for Z ≫ Zcr,1 is demonstrated. The dependence of the Casimir energy on the screening of the Coulomb asymptotics of the external source at some R1 > R0 is also explored in detail, and some peculiar effects in the partial channels with the lowest rotational numbers mj = ±1/2 , ±3/2 in the screened case are also discussed.