Hexagonal boron nitride (hBN) has been incorporated, as an active filler, in a customized silicone matrix to obtain high thermal conductivity composites, maintaining high flexibility and low dielectric permittivity, which are of interest for heat dissipation in energy storage systems (e.g., batteries or supercapacitors) and electronics. By the proper processing of the filler (i.e., hydrophobization with octamethylcyclotetrasiloxane and ultrasonic exfoliation) and its optimal loading (i.e., 10 wt%), composites with thermal conductivity up to 3.543 W·m−1·K−1 were obtained. Conductive heat flow (−280.04 W), measured in real heating–cooling conditions, proved to be superior to that of a commercial heatsink paste (−161.92 W), which has a much higher density (2.5 g/cm3 compared to 1.05 g/cm3 of these composites). The mechanical and electrical properties are also affected in a favorable way (increased modulus and elongation, low dielectric losses, and electrical conductivity) for applications as thermal management materials.