This paper reports on a search for flare emission via charge-exchange radiation in the wings of the Lyman-alpha line of He ii at 304 A, as originally suggested for hydrogen by Orrall and Zirker. Via this mechanism a primary alpha particle that penetrates into the neutral chromosphere can pick up an atomic electron and emit in the He ii bound-bound spectrum before it stops. The Extreme-ultraviolet Variability Experiment (EVE) on board the Solar Dynamics Observatory (SDO) gives us our first chance to search for this effect systematically. The Orrall-Zirker mechanism has great importance for flare physics because of the essential roles that particle acceleration plays; this mechanism is one of the few proposed that would allow remote sensing of primary accelerated particles below a few MeV/nucleon. We study ten events in total, including the gamma-ray events SOL2010-06-12 (M2.0) and SOL2011-02-24 (M3.5) (the latter a limb flare), seven X-class flares, and one prominent M-class event that produced solar energetic particles (SEPs). The absence of charge-exchange line wings may point to a need for more complete theoretical work. Some of the events do have broad-band signatures, which could correspond to continua from other origins, but these do not have the spectral signatures expected from the Orrall-Zirker mechanism.