We present the results of our initial survey of H II galaxies searching for neutral hydrogen companion objects. In our sample of nine galaxies, we detect nearby H I companions for four. Here we present the H I observations of those four galaxies and their companions. The galaxies range in H I mass from 2×108 to 7×109 script M sign⊙, and in total mass from 6×109 to 7×1010 script M sign⊙. The masses of the companions tend to be an order of magnitude less than those of their parent galaxy in both H I and total mass. In no cases did the mass of the companion approach that of the H II galaxy. We find that the probability of an H II galaxy having a companion must be at least 0.17. Any lower than this and a detection rate of 4/9 would be extremely unlikely in an experiment such as ours. This detection of companions around apparently isolated H II galaxies supports our hypothesis that galaxies experiencing a burst of star formation are good places to search for small H I rich companions. The companions span the same range in mass and column density as the population of Lya absorbers seen in the spectra of QSOs and may form a low redshift analog to that population. Companions such as these, less massive than the parent galaxy, may be fragments left over from the epoch of galaxy formation which escaped incorporation into the galaxy. We compare our sample with a sample of H II galaxy/companion systems from the literature and find that half of each sample seem to have bars.