The nearby star-forming dwarf galaxy Sextans A offers a unique window into galaxy evolution in the early Universe, owing to its extremely low metallicity (∼0.01─0.07 Z⊙). Recent JWST imaging of Sextans A spanning 1─21 μm enables a detailed characterization of its dusty stellar populations and interstellar medium. In this work, we compare the observed JWST color─magnitude distributions of evolved stars with stellar evolution and dust-formation models to characterize the properties of the asymptotic giant branch (AGB) population, including progenitor mass, formation epoch, metallicity, and dust production. Evolutionary tracks for 0.8─7 M⊙ stars with metallicity Z = 10−3 provide good agreement with the overall distribution of AGB stars in Sextans A. More than 90% of the AGB population occupies a nearly vertical sequence in the color─magnitude diagrams, corresponding to stars spanning a wide range of masses and ages but exhibiting little or no circumstellar dust. This sequence appears to be dominated by oxygen-rich (M-type) AGB stars and reveals that the F444W flux is a robust luminosity diagnostic. A small subset of sources displays strong IR excesses and is dominated by carbon stars descending from 1.25 to 1.5 M⊙ progenitors that formed ∼2─3 Gyr ago and are currently in the final AGB phases. Their MIRI colors imply very low metallicities, consistent with estimates from the red giant branch morphology (∼0.01─0.02 Z⊙). Finally, we show that the JWST/NIRCam F277W ─ F444W color serves as an effective proxy for the dust production rate, with models predicting rates up to ∼10−7 M⊙ yr−1 for the reddest sources in Sextans A.