The flux of cosmic-ray antiprotons from neutralino annihilations in the Galactic halo is computed for a large sample of models in the minimal supersymmetric extension of the standard model (MSSM). We also revisit the problem of estimating the background of low-energy cosmic-ray-induced secondary antiprotons, taking into account their subsequent interactions (and energy loss) and the presence of nuclei in the interstellar matter. We consider a two-zone diffusion model, with and without a Galactic wind. We find that, given the uncertainties in the background predictions, there is no need for a primary (exotic) component to explain current data. However, allowing for a signal by playing with the uncertainties in the background estimate, we discuss the characteristic features of the supersymmetric models that give a satisfactory description of the data. We point out that in some cases, the optimal kinetic energy to search for a signal from supersymmetric dark matter is above several GeV, rather than the traditional sub-GeV region. The large astrophysical uncertainties involved do not, on the other hand, allow the exclusion of any of the MSSM models we consider on the basis of data. In addition to numerical results, we also present convenient parameterizations of the antiproton yields of all "basic" two-body final states. We also give examples of the yield and differential energy spectrum for a set of supersymmetric models with high rates. We also remark that it is difficult to set a limit on the antiproton lifetime from present measurements, since the injection of antiprotons from neutralino annihilation can compensate for the loss from decay.