Weighing in since I've been rinsing my nose 2x day for many years. I have used previously boiled water only, which I store in the fridge. Every 5 days I boil another batch. NeilMed specifically says to use boiled water.

My understanding about distilled water is that although the minerals have been removed from it, it is not necessarily sterile. The container may have been contaminated before filling. And once a gallon container is opened it could also become contaminated and therefore not sterile.