We perform a census of the reddest, and potentially youngest, protostars in the Orion molecular clouds using data obtained with the PACS instrument on board the Herschel Space Observatory and the LABOCA and SABOCA instruments on APEX as part of the Herschel Orion Protostar Survey (HOPS). A total of 55 new protostar candidates are detected at 70 μm and 160 μm that are either too faint (m 24 > 7 mag) to be reliably classified as protostars or undetected in the Spitzer/MIPS 24 μm band. We find that the 11 reddest protostar candidates with log λF λ 70/λF λ 24 > 1.65 are free of contamination and can thus be reliably explained as protostars. The remaining 44 sources have less extreme 70/24 colors, fainter 70 μm fluxes, and higher levels of contamination. Taking the previously known sample of Spitzer protostars and the new sample together, we find 18 sources that have log λF λ 70/λF λ 24 > 1.65; we name these sources "PACS Bright Red sources," or PBRs. Our analysis reveals that the PBR sample is composed of Class 0 like sources characterized by very red spectral energy distributions (SEDs; T bol < 45 K) and large values of sub-millimeter fluxes (L smm /L bol > 0.6%). Modified blackbody fits to the SEDs provide lower limits to the envelope masses of 0.2-2 M and luminosities of 0.7-10 L . Based on these properties, and a comparison of the SEDs with radiative transfer models of protostars, we conclude that the PBRs are most likely extreme Class 0 objects distinguished by higher than typical envelope densities and hence, high mass infall rates.