Variation of double slits experiment Setup goes something like this: the laser gun fires only 1 photon each time and the only way for the photon to appear on the hidden screen is for them to be reflected from the 2 narrow mirrors.(see image below)

I was watching a ping pong match and suddenly this pops into my mind.
Will there be any interference pattern based on my setup?
I argue that 1 photon now does not have the chance to interfere with itself like the double slits so there will not be any zebra pattern showing up but I might be wrong. Also if I coat both mirrors with Polaroid so that one mirror is left circularly polarized while the other is right circularly polarized, what will appears on the hidden screen if any?
 A: As an experimental physicist I would advice you to do the experiment.
What the theory predicts for single photons is what the boundary conditions the wavefunction of the photon has obeyed for the particular experiment. This wavefunction is complex and carries the phase information for building up the classical electromagnetic wave. It should not be surprising because both the classical wave and the photons it is composed of are solutions of the same maxwell equations, in the case of the photon treated as operators on the wavefunction.
Thus, if interference is seen in a classical light experiment,the single photon distributions will build up to the interference pattern. The classical em distribution is the probability density of finding a photon at a screen, and thus it is the square of the wavefunction of the individual photon. For links look at this answer of mine.
A: There is perhaps a bit more that we can add to the answer of anna v in the context of an experiment done with single photons. How would one do such an experiment? First one would need to do the alignment. One would set up the light source(s) so that no light falls on the screen directly. It would be important that the light is coherent so that one should in principle be able to see interference if there is an overlap. Then one would place the mirrors for the light to be reflected onto the screen so that the two reflected beams overlap on the screen. To do this alignment, one would first use a bright source that one can see - classical light. If the light is coherence one should see interference on the screen. 
Now the setup is done and one can turn to the single photon case. So in some way one modifies the light source (not important how) so that it emits one photon at a time. (Just to be more specific what we mean by this: the light intensity is low enough so that the photons are registered individually by the detector.) The screen needs to be replace by a detector array that can detect single photons (they do exist). We then wait for a while to accumulate enough detections of single photons to see if the resulting pattern shows an interference pattern.
What will we see? Well, as anna v explained, if we saw interference with the classical light we will also see interference in the case of single photons. Why? Because, the theory that describes the classical case and the theory that describes the quantum case for single photons in a linear system are identical. These theories have been tested quite thoroughly so we trust them.
