We present an analysis of elemental abundances of He, N, O, Ne, S, and Ar in Magellanic Cloud planetary nebulae (PNe), and focus initially on 14 PNe in the Small Magellanic Cloud (SMC). We derived the abundances from a combination of deep, high dispersion optical spectra, as well as mid-infrared (IR) spectra from the Spitzer Space Telescope. A detailed comparison with prior SMC PN studies shows that significant variations among authors of relative emission line flux determinations lead to systematic discrepancies in derived elemental abundances between studies that are >~0.15 dex, in spite of similar analysis methods. We used ionic abundances derived from IR emission lines, including those from ionization stages not observable in the optical, to examine the accuracy of some commonly used recipes for ionization correction factors (ICFs). These ICFs, which were developed for ions observed in the optical and ultraviolet, relate ionic abundances to total elemental abundances. We find that most of these ICFs work very well even in the limit of substantially sub-Solar metallicities, except for PNe with very high ionization. Our abundance analysis shows enhancements of He and N that are predicted from prior dredge-up processes of the progenitors on the AGB, as well as the well known correlations among O, Ne, S, and Ar that are little affected by nucleosynthesis in this mass range. We identified MG_8 as an interesting limiting case of a PN central star with a ~3.5 M_sun progenitor in which hot-bottom burning did not occur in its prior AGB evolution. We find no evidence for O depletion in the progenitor AGB stars via the O-N cycle, which is consistent with predictions for lower-mass stars. We also find low S/O ratios relative to SMC H_II regions, with a deficit comparable to what has been found for Galactic PNe.