Astronauts often see flashes when they are in space. How often these particles make their ways down to the Earth and hit us? What's the chance they hit our eyes and see flashes?
Muons are the main cosmic rays reaching sea level with a possibility of interacting with matter. Neutrinos interact very weakly and need special detectors to be seen at all.
Rule of thumb when I was working with counters is that the flux of muons at sea level is 1 per cm^2 per second, wikipedia gives this number for over 1 Gev
For energies above 1 GeV see this link.
The question then becomes biophysical, can the eye detect ionizing deposition which lasts much less than a second as the velocities are very high? The answer is no, unless there is an interaction with nuclei in the eye and there is a shower in the eye. The flux at sea level is predominantly muons which interact weakly with nuclei so there is very small probability that a shine in the dark is a cosmic interaction.
In the stratosphere and where the astronauts are living the flux is composed mainly by hadrons, and those have a high probability of interacting with nuclei in the eye.