The paradigm that gamma-ray burst (GRB) fireballs are the sources of the ultra-high energy cosmic rays (UHECRs) is being probed by neutrino observations. Very stringent bounds can be obtained from the cosmic ray (proton)--neutrino connection, assuming that the UHECRs escape as neutrons. In this study, we identify three different regimes as a function of the fireball parameters: the standard "one neutrino per cosmic ray" case, the optically thick (to neutron escape) case, and the case where leakage of protons from the boundaries of the shells (direct escape) dominates. In the optically thick regime, photomeson production is very efficient, and more neutrinos will be emitted per cosmic ray than in the standard case, whereas in the direct escape-dominated regime, more cosmic rays than neutrinos will be emitted. We demonstrate that, for efficient proton acceleration, which is required to describe the observed UHECR spectrum, the standard case only applies to a very narrow region of the fireball parameter space. We illustrate with several observed examples that conclusions on the cosmic ray--neutrino connection will depend on the actual burst parameters. We show that the definition of the pion production efficiency currently used by the IceCube collaboration underestimates the neutrino production in the optically thick case. Finally, we point out that the direct escape component leads to a spectral break in the cosmic ray spectrum emitted from a single source. The resulting "two-component model" can be used to even more strongly pronounce the spectral features of the observed UHECR spectrum than the dip model.