We propose a general, easy-to-implement scheme for broadband coherent perfect absorption (CPA) using epsilon-near-zero (ENZ) multilayer films. Specifically, we employ indium tin oxide (ITO) as a tunable ENZ material, and theoretically investigate CPA in the near-infrared region. We first derive general CPA conditions using the scattering matrix and the admittance matching methods. Then, by combining these two methods, we extract analytic expressions for all relevant parameters for CPA. Based on this theoretical framework, we proceed to study ENZ CPA in a single layer ITO film and apply it to all-optical switching. Finally, using an ITO multilayer of different ENZ wavelengths, we implement broadband ENZ CPA structures and investigate multi-wavelength all-optical switching in the technologically important telecommunication window. In our design, the admittance matching diagram was employed to graphically extract not only the structural parameters (the film thicknesses and incident angles), but also the input beam parameters (the irradiance ratio and phase difference between two input beams). We find that the multi-wavelength all-optical switching in our broadband ENZ CPA system can be fully controlled by the phase difference between two input beams. The simple but general design principles and analyses in this work can be widely used in various thin-film devices.