Normal civilian reactors work on low enrichment uranium or even natural uranium--stuff that has no potential to go boom.

Even reprocessing isn't the danger it's made out to be. First, the plutonium from spent reactor fuel has a lot of Pu-240 in it. Bombs need Pu-239, too much Pu-240 will make them malfunction. (If you are trying to make Pu-239 you switch out the fuel rods much more frequently.) Second, the reprocessing plant has access to a lot of very hot stuff. All you actually need to do in reprocessing is strip out the waste products that poison the reaction, a fuel rod heavily "contaminated" with something like Cobalt-60 won't interfere with reactor operation, but it will ensure no thief will make off with it.

Naval reactors, however, are built to be as small as possible. That means very highly enriched uranium. Building a gun-type uranium bomb is easily within the range of what Al Qaeda can do, the limiting factor is obtaining the materials. Thus naval reactor fuel needs to be treated with extreme security.