JWST has revealed diverse new populations of high-redshift ($z\sim4-11$) AGN and extreme star-forming galaxies that challenge current models. In this paper, we use rest-frame UV emission-line diagnostics to identify AGN candidates and other exceptional ionizing sources, complementing previous studies predominantly focused on broad-line AGN. In this paper, we use rest-frame UV emission-line diagnostics to identify AGN candidates and other exceptional ionizing sources, complementing previous studies predominantly focused on broad-line AGN. From a parent sample of 205 $\mathrm{z_{spec}}>3$ UNCOVER galaxies with NIRSpec/PRISM follow-up, we identify 12 C IV, He II, and C III] emitters. Leveraging the combined rest-optical and UV coverage of PRISM, we limit the emission-line model space using the sample's [O III]/H$\beta$ distribution, significantly decreasing the overlap between AGN and star-formation models in the UV diagnostics. We then find that the five He II emitters are the strongest AGN candidates, with further support from two [Ne V] detections and one X-ray detection from Chandra. We cannot robustly quantify the AGN fraction in this sample, but we note that close to 20% of $\mathrm{M_{*}>2\times10^{9}\,M_{\odot}}$ parent sample galaxies are AGN candidates. The lower-mass line emitters, which are consistent with both AGN and star-forming photoionization models, have more compact sizes and higher specific star formation rates than the parent sample. Higher-resolution and deeper data on these UV line emitters should provide much stronger constraints on the obscured AGN fraction at $z > 3$.