We investigate dust production and stellar mass loss in the Galactic globular cluster NGC 362. Due to its close proximity to the Small Magellanic Cloud (SMC), NGC 362 was imaged with the IRAC and MIPS cameras onboard the Spitzer Space Telescope as part of the Surveying the Agents of Galaxy Evolution (SAGE-SMC) Spitzer Legacy program. We detect several cluster members near the tip of the Red Giant Branch that exhibit infrared excesses indicative of circumstellar dust and find that dust is not present in measurable quantities in stars below the tip of the Red Giant Branch. We modeled the spectral energy distribution (SED) of the stars with the strongest IR excess and find a total cluster dust mass-loss rate of 3.0(+2.0/-1.2) x 10^-9 solar masses per year, corresponding to a gas mass-loss rate of 8.6(+5.6/-3.4) x 10^-6 solar masses per year, assuming [Fe/H] = -1.16. This mass loss is in addition to any dust-less mass loss that is certainly occurring within the cluster. The two most extreme stars, variables V2 and V16, contribute up to 45% of the total cluster dust-traced mass loss. The SEDs of the more moderate stars indicate the presence of silicate dust, as expected for low-mass, low-metallicity stars. Surprisingly, the SED shapes of the stars with the strongest mass-loss rates appear to require the presence of amorphous carbon dust, possibly in combination with silicate dust, despite their oxygen-rich nature. These results corroborate our previous findings in omega Centauri.