Ryuk said that about users of the Death Note not going to either. I don't know what it means when humans go into nothingness. It could mean that they're reincarnated, which would cause them to forget their lives and their memories to go into nothingness.

Yes, but when Light said there wasn't a Heaven or Hell for anyone to go to, Ryuk basically confirmed it. He wouldn't have any reason to lie to Light, just as he had no reason to tell Light anything about Raye Penber. Also, when the kid who had Ryuk's death note briefly died, Ryuk implied that, at least the Death Note holders, went to a realm of the dead, maybe something like in Egyptian mythology.

One must remember that Death Note has a Japanese writer, and many people in Japan are Buddhist, or are at least familiar with Buddhist ideas and traditions. In Buddhism, you're reincarnated after you die, so it wouldn't be a surprise for this to happen in the world of Death Note.

I imagine that whatever religion is central to Death Note is whichever one mentions shinigami. Are there shinigami in buddhism or shinto? I do believe shinto is still a major influence in Japan.

A shinigami is the Japanese version of the Grim Reaper. It's not really part of Shintoism, it's really a concept that they got from the Europeans (I think).