The nature of the close-in rocky planet 55 Cnc e is puzzling despite having been observed extensively. Its optical and infrared occultation depths show temporal variability, in addition to a phase curve variability observed in the optical. We wish to explore the possibility that the variability originates from the planet being in a 3:2 spin-orbit resonance, thus showing different sides during occultations. We proposed and were awarded Cycle 1 time at the James Webb Space Telescope (JWST) to test this hypothesis. JWST/NIRCam (Near Infrared Camera) observed five occultations (secondary eclipses), of which four were observed within a week, of the planet simultaneously at 2.1 and 4.5um. While the former gives band-integrated photometry, the latter provides a spectrum between 3.9-5.0um. We find that the occultation depths in both bandpasses are highly variable and change between a non-detection (-5+/-6ppm and 7+/-9ppm) to 96+/-8ppm and 119^+34^_-19_ppm at 2.1um and 4.5um, respectively. Interestingly, the variations in both bandpasses are not correlated and do not support the 3:2 spin-orbit resonance explanation. The measured brightness temperature at 4.5um varies between 873-2256K and is lower than the expected dayside temperature of bare rock with no heat re-distribution (2500K) which is indicative of an atmosphere. Our atmospheric retrieval analysis of occultation depth spectra at 4.5um finds that different visits statistically favour various atmospheric scenarios including a thin outgassed CO/CO2 atmosphere and a silicate rock vapour atmosphere. Some visits even support a flat line model. The observed variability could be explained by stochastic outgassing of CO/CO2, which is also hinted by retrievals. Alternatively, the variability, observed at both 2.1 and 4.5um, could be the result of a circumstellar patchy dust torus generated by volcanism on the planet.