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.

10 pages, 5 figures, A&A accepted