We present near-infrared JHK_S_ light curves for the double-lined eclipsing binary system Two Micron All Sky Survey J05352184-0546085, in which both components have been shown to be brown dwarfs with an age of ~1Myr. We analyze these light curves together with the previously published I_C_-band light curve and radial velocities to provide refined measurements of the system's physical parameters. The component masses and radii are here determined with an accuracy of ~6.5% and ~1.5%, respectively. In addition, we confirm the previous surprising finding that the primary brown dwarf has a cooler effective temperature than its lower mass companion. Our analysis reveals two low-amplitude (~0.02mag) periodic signals, one attributable to the rotation of the primary with a period of 3.293+/-0.001d and the other to the rotation of the secondary with a period of 14.05+/-0.05d. Both periods are consistent with the measured vsin i and radii.