This paper analyzes a sample of 489 Spitzer/Infrared Array Camera (IRAC) sources in the Extended Groth Strip (EGS), whose spectral energy distributions fit a red power law (PL) from 3.6 to 8.0um. The median redshift for sources with known redshifts is <z>=1.6. Though all or nearly all of the sample galaxies are likely to be active galactic nuclei (AGNs), only 33% were detected in the EGS X-ray survey (AEGIS-X) using 200ks Chandra observations. The detected sources are X-ray luminous with L_X_>10^43^erg/s and moderately to heavily obscured with N_H_>10^22^cm^-2^. Stacking the X-ray-undetected sample members yields a statistically significant X-ray signal, suggesting that they are on average more distant or more obscured than sources with X-ray detections. The ratio of X-ray to mid-infrared fluxes suggests that a substantial fraction of the sources undetected in X-rays are obscured at the Compton-thick level, in contrast to the X-ray-detected sources, all of which appear to be Compton thin.