This isn't entirely correct. Distilled water is fine to drink and the majority of our freshwater water is highly hypo-osmotic relative to our blood (that is why blood tastes salty and drinking water doesn't). While our body fluids contain many minerals, it is really sodium (Na+) that constitutes the major ion regulating blood osmolarity.
Your logic is generally correct, that replacing blood volume with large volumes of hypoosmotic water will draw salts from your cells. However, we do not drink large volumes of water relative to our total body water content. Our bodies are made of 80% water with about 0.9% salt content. For a typical 70 kg individual, that represents 56 kg or 56 L of water and about 500 g of salt. At most we can only drink at a sitting between 2-3 L of water. Lets say you drank 3L of pure water, your body would have to supply this water with 27g of salt to make it isoosmotic with your blood. This reflects only 5% of the total body salt content which is hardly a difficulty for the homeostatic mechanisms of your body.
Besides, having not enough salts is not a very common problem as our diets, and in the case of the show (saltwater everywhere), has an excess of minerals. We obtain most of our salt from our diet, one reason why deer are attracted to salt licks because their predominately herbivorous food can be lacking in salt. Incidental ingestion of salt by the stars of the the show, even by virtue of breathing in salt-spray from the air, would more then adequately replace their body salt content. Plus their diet, including sea cucumber would furnish an excess of salts and minerals which would be flushed out by the kidneys.
Main point - don't be afraid of drinking distilled water.
