The ubiquity of Lyman alpha (Ly ) emission in a sample of four bright O iii-strong star-forming galaxies with redshifts above seven has led to the suggestion that such luminous sources represent a distinct population compared with their fainter, more numerous counterparts. The presence of Ly emission within the reionization era could indicate that these sources created early ionized bubbles due to their unusually strong radiation, possibly because of the presence of active galactic nuclei. To test this hypothesis, we secured long integration spectra with XSHOOTER on the VLT for three sources selected to have similar luminosities and prominent excess fluxes in the IRAC 3.6 or 4.5 m band, usually attributed to strong O iii emission. We secured additional spectroscopy for one of these galaxies at z = 7.15 using MOSFIRE at the Keck telescope. For the most well-studied source in our sample with the strongest IRAC excess, we detect significant nebular emission from He ii and N v indicative of a non-thermal source. For the other two sources at z = 6.81 and z = 6.85, for which no previous optical/near-infrared spectroscopy was available, Ly is seen in one and C iii emission in the other. Although based on a modest sample, our results further support the hypothesis that the phenomenon of intense O iii emission is associated preferentially with sources lying in early ionized bubbles. However, even though one of our sources at z = 7.15 suggests the presence of non-thermal radiation, such ionized bubbles may not uniquely arise in this manner. We discuss the unique advantages of extending such challenging diagnostic studies with JWST.