Wow, check out how to make HOT ICE! This is awesome if you have some sodium acetate laying around.
A heat pack like the one you are describing contains sodium acetate and water. It turns out that sodium acetate is very good at supercooling. It “freezes” at 130 degrees F (54 degrees C), but it is happy to exist as a liquid at a much lower temperature and is extremely stable. Clicking the disk, however, has the ability to force a few molecules to flip to the solid state, and the rest of the liquid then rushes to solidify as well. The temperature of the solidifying liquid jumps up to 130 degrees F in the process.