  (Abridged) We have used the VLA to search for neutral atomic hydrogen in the circumstellar envelopes of five AGB stars. We have detected HI 21-cm emission coincident in both position and velocity with the semi-regular variable RS Cnc. The emission comprises a compact, slightly elongated feature centered on the star with a mean diameter ~82'' (1.5e17 cm), plus an additional filament extending ~6' to the NW. This filament suggests that a portion of the mass loss is highly asymmetric. We estimate MHI=1.5e-3 Msun and M_dot~1.7e-7 Msun/yr. Toward R Cas, we detect weak emission that peaks at the stellar systemic velocity and overlaps with the location of its circumstellar dust shell and thus is probably related to the star. In the case of IRC+10216, we were unable to confirm the detection of HI in absorption against the cosmic background previously reported by Le Bertre & Gerard. However, we detect arcs of emission at projected distances of r~14'-18' (~2e18 cm) to the NW. A large separation of the emission from the star is plausible given its advanced evolutionary status, although it is unclear if the asymmetric distribution and complex velocity structure are consistent with a circumstellar origin. For EP Aqr, we detected HI emission comprising multiple clumps redward of the systemic velocity, but we are unable to determine unambiguously whether the emission arises from the circumstellar envelope or from interstellar clouds along the line-of-sight. Regardless of the adopted distance for the clumps, their inferred HI masses are at least an order of magnitude smaller than their individual binding masses. We detected our fifth target, R Aqr (a symbiotic binary), in the 1.4 GHz continuum, but did not detect any HI emission from the system. 
VLA Observations of HI in the Circumstellar Envelopes of Asymptotic   Giant Branch Stars