Based on H I 21 cm line and radio continuum observations performed with the Very Large Array of the National Radio Astronomy Observatory and on optical narrowband images obtained at the Cerro Tololo Inter-American Observatory, we investigate the distribution of the ionized and neutral gas in the region of the ring nebula NGC 2359 surrounding the Wolf-Rayet star HD 56925 at a distance of 5 kpc. The radio continuum data, obtained at 1465 MHz with an angular resolution of ~30", have the same morphology as the optical Hα nebular line emission. An ionized mass of 70 M_solar and an electron density (corrected for filling factor) of 120 cm^-3 were estimated for the filamentary shell, while the mass in the whole ionized complex is in the range 850-1100 M_solar. The H I 21 cm line data, with angular and velocity resolutions of ~45" and 1.3 km s^-1, respectively, reveal for the first time neutral hydrogen clearly interacting with the optical ring nebula at its systemic velocity of 54 km s^-1 and at 63 km s^-1. The H I gas distribution at 54 km s^-1 indicates the location of the ionization fronts. The estimated total neutral mass is ~320 M_solar. Most of this mass seems to be located in a hemispherical shell expanding at 6-7 km s^-1. Taking into account the ionized, neutral, and molecular components, the mass associated with the NGC 2359 complex is about 2200 M_solar. The dynamics of the nebula are consistent with momentum conservation or with an intermediate stage between energy and momentum conservation.