We analyze a sample of 25 [Ne v] (lambda 3426) emission-line galaxies at 1.4 < z < 2.3 using Hubble Space Telescope/Wide Field Camera 3 G102 and G141 grism observations from the CANDELS Ly alpha Emission at Reionization (CLEAR) survey. [Ne v] emission probes extremely energetic photoionization (creation potential of 97.11 eV) and is often attributed to energetic radiation from active galactic nuclei (AGNs), shocks from supernovae, or an otherwise very hard ionizing spectrum from the stellar continuum. In this work, we use [Ne v] in conjunction with other rest-frame UV/optical emission lines ([O ii] lambda lambda 3726, 3729, [Ne iii] lambda 3869, H beta, [O iii] lambda lambda 4959, 5007, H alpha+[N ii] lambda lambda 6548, 6583, [S ii] lambda lambda 6716, 6731), deep (2-7 Ms) X-ray observations (from Chandra), and mid-infrared imaging (from Spitzer) to study the origin of this emission and to place constraints on the nature of the ionizing engine. The majority of the [Ne v]-detected galaxies have properties consistent with ionization from AGNs. However, for our [Ne v]-selected sample, the X-ray luminosities are consistent with local (z less than or similar to 0.1) X-ray-selected Seyferts, but the [Ne v] luminosities are more consistent with those from z similar to 1 X-ray-selected QSOs. The excess [Ne v] emission requires either reduced hard X-rays or a similar to 0.1 keV excess. We discuss possible origins of the apparent [Ne v] excess, which could be related to the "soft (X-ray) excess" observed in some QSOs and Seyferts and/or be a consequence of a complex/anisotropic geometry for the narrow-line region, combined with absorption from a warm, relativistic wind ejected from the accretion disk. We also consider implications for future studies of extreme high-ionization systems in the epoch of reionization (z greater than or similar to 6) with the James Webb Space Telescope.