We have used the Infrared Spectrograph (IRS) on the Spitzer Space Telescope to observe stars in the Small Magellanic Cloud (SMC) selected from the Point Source Catalog of the Midcourse Space Experiment (MSX). We concentrate on the dust properties of the oxygen-rich evolved stars. The dust composition has smaller contributions from alumina compared to the Galaxy. This difference may arise from the lower metallicity in the SMC, but it could be a selection effect, as the SMC sample includes more stars that are brighter and thus more massive. The distribution of the SMC stars along the silicate sequence looks more like the Galactic sample of red supergiants than asymptotic giant branch stars (AGBs). While many of the SMC stars are definitively on the AGB, several also show evidence of hot bottom burning. Three of the supergiants show PAH emission at 11.3{mu}m. Two other sources show mixed chemistry, with both carbon-rich and oxygen-rich spectral features. One, MSX SMC 134, may be the first confirmed silicate/carbon star in the SMC. The other, MSX SMC 049, is a candidate post-AGB star. MSX SMC 145, previously considered a candidate OH/IR star, is actually an AGB star with a background galaxy at z=0.16 along the same line of sight. We consider the overall characteristics of all the MSX sources, the most infrared-bright objects in the SMC, in light of the higher sensitivity and resolution of Spitzer, and compare them with the object types expected from the original selection criteria. This population represents what will be seen in more distant galaxies by the upcoming James Webb Space Telescope (JWST). Color-color diagrams generated from the IRS spectra and the mid-infrared filters on JWST show how one can separate evolved stars from young stellar objects (YSOs) and distinguish among different classes of YSOs.