Void population consists mainly of late-type and low surface brightness (LSB) dwarf galaxies, whose atomic hydrogen is the main component of their baryonic matter. Therefore observations of void galaxy HI are mandatory to understand both their evolution and dynamics. Our aim was to obtain integrated HI parameters for a fainter part of the nearby Lynx-Cancer void galaxy sample (total of 45 objects) with the Nancay Radio Telescope (NRT) and to conduct the comparative analysis of the whole 103 void galaxies with known HI data with a sample of similar galaxies residing in denser environments of the Local Volume. For HI observations we used the NRT with its sensitive antenna/receiver system FORT and standard processing. The comparison of the void and "control" samples on the parameter M(HI)/L_B_ is conducted with the non-parametric method "The 2x2 Contingency Table test". We obtained new HI data for about 40% of the Lynx-Cancer galaxy sample. Along with data from the literature, we use for further analysis data for 103 void objects. The proxy of the evolutional parameter M(HI)/L_B_ of the void sample is compared with that of 82 galaxies of morphological types 8-10 residing in the Local Volume groups and aggregates. At the confidence level of P=0.988, we conclude that for the same luminosity, these void galaxies are systematically gas-richer, in average by ~39%. This result is consistent with the authors' earlier conclusion on the smaller gas metallicities and evidences for the slower low-mass galaxy evolution in voids.