We present neutral hydrogen (HI) observations using the Robert C. Byrd Green Bank Telescope (GBT) of 70 optically detected UDG candidates in the Coma region from the Systematically Measuring Ultra-Diffuse Galaxies survey (SMUDGes). We detect HI in 18 targets, confirming nine to be gas-rich UDGs and the remainder to be foreground dwarfs. None of our HI-detected UDGs are Coma Cluster members and all but one are in low-density environments. The HI-detected UDGs are bluer and have more irregular morphologies than the redder, smoother candidates not detected in HI, with the combination of optical color and morphology being a better predictor of gas richness than either parameter alone. There is little visual difference between the gas-rich UDGs and the foreground dwarfs in the SMUDGes imaging, and distances are needed to distinguish between them. We find that the gas richnesses of our HI-confirmed UDGs and those from other samples scale with their effective radii in two stellar mass bins, possibly providing clues to their formation. We attempt to place our UDGs on the baryonic Tully-Fisher relation (BTFR) using optical ellipticities and turbulence-corrected HI line widths to estimate rotation velocities, but the potential systematics associated with fitting smooth Sersic profiles to clumpy, low-inclination disks of low surface brightness precludes a meaningful analysis of potential BTFR offsets. These observations are a pilot for a large campaign now under way at the GBT to use the HI properties of gas-rich UDGs to quantitatively constrain how these galaxies form and evolve.