Instant cold packs contain ammonium nitrate and a sealed packet of water. When ammonium nitrate dissolves in water, it cools down a lot. Breaking the packet allows the water and ammonium nitrate to mix.
Ammonium nitrate is slightly toxic, as are all nitrates, but not too bad. It is very widely used as a high-nitrogen fertilizer and as an oxidizer in explosives. It is put to use in the environment very quickly, and does not persist long.
Nitrates in water are an environmental hazard - they can trigger algae blooms, and when these blooms die and decompose, oxygen levels in the water drop. The resulting "dead zone" is unable to support fish and crustaceans. However, the couple of grams of ammonium nitrate in cold packs are nothing compared to the tons and tons of nitrates applied to farms. Agricultural runoff is the main source of nitrogen pollution in water.
The plastic parts of the cold pack will take a long time to degrade, and are difficult to recycle. These are probably the least-environmentally-friendly part of the cold packs.
So the pros of this cold pack design are that the cooling effect is available instantly on-demand (as opposed to a pack that needs to be kept in a freezer), and that the ingredients are cheap and fairly non-toxic. The cons would be that they can only be used once, and that disposal of the pack produces persistent waste.