  We study the proposal that the cosmic ray primaries above the
Greisen-Zatsepin-Kuzmin (GZK) cutoff are gluino-containing hadrons ($\tilde
g$-hadrons). We describe the interaction of $\tilde g$-hadrons with nucleons in
the framework of the Gribov-Regge approach using a modified version of the
hadronic interaction model QGSJET for the generations of Extensive Air Showers
(EAS). There are two mass windows marginally allowed for gluinos: $m_{\tilde
g}\lsim 3$ GeV and $25\lsim m_{\tilde g}\lsim 35$ GeV. Gluino-containing
hadrons corresponding to the second window produce EAS very different from the
observed ones. Light $\tilde g$-hadrons corresponding to the first gluino
window produce EAS similar to those initiated by protons, and only future
detectors can marginally distinguish them. We propose a beam-dump accelerator
experiment to search for $\tilde g$-hadrons in this mass window. We emphasize
the importance of this experiment: it can discover (or exclude) the light
gluino and its role as a cosmic ray primary at ultra high energies.
