We present new Spitzer IRAC/PU/MIPS photometry from 3.6 to 24 μm, and new Gemini GMOS photometry at 0.48 μm, of the young brown dwarf eclipsing binary 2MASS J05352184–0546085, located in the Orion Nebula Cluster. No excess disk emission is detected. The measured fluxes at λ ≤ 8 μm are within 1σ (0.1 mJy) of a bare photosphere, and the 3σ upper limit at 16 μm is a mere 0.04 mJy above the bare photospheric level. Together with the known properties of the system, this implies the absence of optically thick disks around the individual components. It also implies that if any circumbinary disk is present, it must either be optically thin and extremely tenuous (10^–10 M _☉) if it extends in to within ~0.1 AU of the binary (the approximate tidal truncation radius), or it must be optically thick with a large inner hole, >0.6-10 AU in radius depending on degree of flaring. The consequence in all cases is that disk accretion is likely to be negligible or absent. This supports the recent proposal that the strong Hα emission in the primary (more massive) brown dwarf results from chromospheric activity, and thereby bolsters the hypothesis that the surprising T _eff inversion observed between the components is due to strong magnetic fields on the primary. Our data also set constraints on the T _eff of the components independent of spectral type, and thereby on models of the aforementioned magnetic field effects. We discuss the consequences for the derived fundamental properties of young brown dwarfs and very low mass stars in general. Specifically, if very active isolated young brown dwarfs and very low mass stars suffer the same activity/field related effects as the 2M0535–05 primary, the low-mass stellar/substellar initial mass function currently derived from standard evolutionary tracks may be substantially in error.