Controlling the output light-intensity and realizing the light-switch function in hollow-core anti-resonant fibers (HC-ARFs) is crucial for their applications in polarizers, lasers, and sensor systems. Here, we theoretically propose a hybrid light-intensity-tunable HC-ARF deposited with the sandwiched graphene/hexagonal boron nitride/graphene based on the typical six-circular-tube and the nested structures. Changing the external drive voltage from 12.3 to 31.8 V, the hybrid HC-ARF experiences a high-low alterative attenuation coefficient with a modulation depth 3.87 and 1.91 dB/cm for the six-circular-tube and nested structures respectively, serving as a well-performance light-switch at the optical communication wavelength of 1.55 µm. This response is attributed to the variation of the Fermi level of graphene and is obviously influenced by the core size, fiber length, and the number of graphene and hBN layers. Moreover, one attenuation dip of the modulation depth was found because of the epsilon-near-zero effect in graphene. Our design provides a feasible paradigm for integrating graphene with anti-resonant fibers and high-performance electro-optic modulators.