There definitely truth to their response:
Used fuel is stored underwater in a stainless steel lined concrete pool for at least 5 years.
The above statement is a Nuclear Regulatory Commission (NRC) requirement. The NRC is the federal group responsible for licensing nuclear reactors within the United States.
The facility design includes on-site dry cask storage of the used fuel for the life of the facility, or until such time that the used fuel can be ultimately stored at a permanent long-term used fuel repository.
When spent fuel rods meet the time and power requirement for removal from the pool, they get placed into dry cask storage. I've been involved in this operation multiple times now. It's pretty amazing. They place used fuel within an intermediate container (placed inside of the pool) and lift it filled with water to a separate area.
They then pump out the water from the intermediate container with Helium and completely dry it out. Once that's done, they transfer it to the dry cask storage area and transfer the fuel from the intermediate container to a more robust dry cask (called an overpack) for storage. The reduction in radiation levels alone is staggering. From radiation levels that could kill you in seconds to just slightly above background alone.
Here's a link to a dry cask storage process if you're interested:
What's interesting is that used nuclear fuel still has considerable energy available, and could be recovered. There are existing and new technologies on the drawing boards today that can utilize this fuel.
This statement is somewhat accurate. Yes, there are new and existing technologies, but recycling or reusing nuclear fuel isn't allowed within the United States at this time. The new and existing technologies available are used in countries like France. What I know for certain is that no commercial nuclear plant reuses or recycles fuel in the US. I don't know for certain if there are any experimental reactors here that fulfills that function.