We consider the tunneling current through a double point-contact Fabry-Pérot interferometer such as used in recent experimental studies of the fractional quantum Hall plateau at filling fraction =5/2. We compare the predictions of several different models of the state of the electrons at this plateau: the Moore-Read, anti- Pfaffian, SU22 NAF, K=8 strong pairing, and 3,3,1 states. All of these predict the existence of charge e/4 quasiparticles, but the first three are non-Abelian while the last two are Abelian. We give explicit formulas for the scaling of charge e/2 and charge e/4 quasiparticle contributions to the current as a function of temperature, gate voltage, and distance between the two point contacts for all three models. Based on these, we analyze several possible explanations of two phenomena reported for recent experiments by Willett et al., namely, halving of the period of the observed resistance oscillations with rising temperature and alternation between the same two observed periods at low temperatures as the area of the interference loop is varied with a side gate. We conclude that the most likely explanation is that the observed alternation is due to switching between even and odd numbers of charge e/4 quasiparticles enclosed within the loop as a function of side-gate voltage, which is a clear signature of the presence of non-Abelian anyons. However, there are important features of the data which do not have a simple explanation within this picture. We suggest further experiments which could help rule out some possible scenarios. We make the corresponding predictions for future tunneling and interference experiments at the other observed second Landau level fractional quantum Hall states.