A low-energy medium-luminosity Muon Collider (MC) is being studied as a possible Higgs Factory (HF). Electrons from muon decays will deposit more than 300 kW in superconducting (SC) magnets of the HF collider ring. This imposes significant challenges to SC magnets used in the HF storage ring (SR) and interaction regions (IR). Conceptual designs of SC dipole and quadrupole magnets are described which provide high operating gradient and field in a large aperture to accommodate the large size of muon beams (due to low β*), as well as a cooling system to intercept the large heat deposition from the showers induced by decay electrons. The distribution of heat deposition in the main elements of HF SR lattice requires large-aperture magnets to accommodate thick high-Z absorbers to protect the SC coils. Based on the developed MARS15 model and intensive simulations, a sophisticated protection system from radiation was designed for the collider SR and IR to bring the peak power density in the SC coils well below the quench limit and reduce the dynamic heat deposition in the cold mass of SC magnets by a factor of 100. The radiation protection system consists of tight tungsten masks in the magnet interconnect regions and elliptical tungsten liners in the magnet aperture optimized for each magnet. These elements reduce also the background particle fluxes in the collider detector.