We study a new population of extremely red objects (EROs) recently discovered by the James Webb Space Telescope (JWST) based on their NIRCam colors F277W - F444W > 1.5 mag. We find 37 EROs in the Cosmic Evolution Early Release Science Survey (CEERS) field with F444W < 28 mag and photometric redshifts between 5 < z < 7, with median z=6.9(-1.6)(+1.0) . Surprisingly, despite their red long-wavelength colors, these EROs have blue short-wavelength colors (F150W - F200W similar to 0 mag) indicative of bimodal spectral energy distributions (SEDs) with a red, steep slope in the rest-frame optical, and a blue, flat slope in the rest-frame UV. Moreover, all these EROs are unresolved, point-like sources in all NIRCam bands. We analyze the SEDs of eight of them with MIRI and NIRSpec observations using stellar population models and active galactic nucleus (AGN) templates. We find that dusty galaxies or obscured AGNs provide similarly good SED fits but different stellar properties: massive and dusty, logM(star)/M-circle dot similar to 10 and A(V) greater than or similar to 3 mag, or low mass and obscured, logM(star)/M-circle dot similar to 7.5 and A(V) similar to 0 mag, hosting an obscured quasi-stellar object (QSO). SED modeling does not favor either scenario, but their unresolved sizes are more suggestive of AGNs. If any EROs are confirmed to have logM(star)/M-circle dot greater than or similar to 10.5, it would increase the pre-JWST number density at z > 7 by up to a factor similar to 60. Similarly, if they are QSOs with luminosities in the L-bol > 10(45-46) erg s(-1) range, their number would exceed that of bright blue QSOs by more than three orders of magnitude. Additional photometry at mid-infrared wavelengths will reveal the true nature of the red continuum emission in these EROs and will place this puzzling population in the right context of galaxy evolution.