  The most massive binary system Eta Carinae has been recently established as a
gamma-ray source by the AGILE and Fermi-LAT detectors. The high energy spectrum
of this gamma-ray source is very intriguing. It shows two clear components and
a lack of any evidence of variability with the orbital period of the binary
system. We consider different scenarios for the acceleration of particles (both
electrons and hadrons) and the production of the high energy radiation in the
model of stellar wind collisions within the binary system Eta Carinae with the
aim to explain the gamma-ray observations and predict the behaviour of the
source at very high gamma-ray energies. The gamma-ray spectra calculated in
terms of the specific models are compared with the observations of Eta Carinae,
and the neutrino spectra produced in hadronic models are confronted with the
atmospheric neutrino background and the sensitivity of 1 km$^2$ neutrino
telescope. We show that spectral features can be explained in terms of the
stellar wind collision model between the winds of the companion stars in which
the acceleration of particles occurs on both sides of the double shock
structure. The shocks from the Eta Carinae star and the companion star can
accelerate particles to different energies depending on the different
conditions determined by the parameters of the stars. The lack of strong GeV
gamma-ray variability with the period of the binary system can be also
understood in terms of such a model. We predict that the gamma-ray emission
features at energies above $\sim$100 GeV will show significant variability (or
its lack) depending on the acceleration and interaction scenario of particles
accelerated within the binary system. For the hadronic models we predict the
expected range of neutrino fluxes from the binary system Eta Carinae.
