We investigate the infrared (IR) properties of cool, evolved stars in the Small Magellanic Cloud (SMC), including the red giant branch (RGB) stars and the dust-producing red supergiant (RSG) and asymptotic giant branch (AGB) stars using observations from the Spitzer Space Telescope Legacy program entitled: "Surveying the Agents of Galaxy Evolution in the Tidally-stripped, Low Metallicity SMC", or SAGE-SMC. The survey includes, for the first time, full spatial coverage of the SMC bar, wing, and tail regions at infrared (IR) wavelengths (3.6 - 160 microns). We identify evolved stars using a combination of near-IR and mid-IR photometry and point out a new feature in the mid-IR color-magnitude diagram that may be due to particularly dusty O-rich AGB stars. We find that the RSG and AGB stars each contribute ~20% of the global SMC flux (extended + point-source) at 3.6 microns, which emphasizes the importance of both stellar types to the integrated flux of distant metal-poor galaxies. The equivalent SAGE survey of the higher-metallicity Large Magellanic Cloud (SAGE-LMC) allows us to explore the influence of metallicity on dust production. We find that the SMC RSG stars are less likely to produce a large amount of dust (as indicated by the [3.6]-[8] color). There is a higher fraction of carbon-rich stars in the SMC, and these stars appear to able to reach colors as red as their LMC counterparts, indicating that C-rich dust forms efficiently in both galaxies. A preliminary estimate of the dust production in AGB and RSG stars reveals that the extreme C-rich AGB stars dominate the dust input in both galaxies, and that the O-rich stars may play a larger role in the LMC than in the SMC. ; Comment: Accepted for publication in AJ. 25 pages, 36 figures, Table 4 will be available electronically from AJ