There is strong evidence that a compact object in the Cygnus X-3 binary system produces an intense beam of ultra-high-energy cosmic rays. Here, we examine the effects of such a beam hitting the companion star and of the subsequent production of secondary neutrinos. We consider how high a beam luminosity is allowed and how high a neutrino to γ-ray (ν/γ) ratio can be obtained from such a system. We find a maximum allowable beam luminosity of ~1042erg s-1 for a system consisting of a compact object and a ~1-10 Msolar main-sequence target star. The proton beam must heat a relatively small area of the target star to satisfy observational constraints on the resulting stellar wind. With such a model, a ν/γ flux ratio of ~103 can result from a combination of γ-ray absorption and a large ν/γ duty cycle ratio. We find that the high density of the atmosphere resulting from compression by the beam leads to pion cascading and a neutrino spectrum peaking at 1-10 GeV energies, which may avoid catastrophic heating of the target star through internal ν interactions. The ν flux and duty cycle are predicted to be accordingly reduced in the energy range above 1 TeV available to a deep underwater neutrino detector.