The killswitch has several functions. One is to destroy evidence of an agent when they are killed and damaging any hostiles who are around. Its other function is security, if an agent goes AWOL then someone at UNATCO can activate the killswitch, and as a result eventually kills the agent. Mechanically augmented agents have a killphrase, similar to the killswitch however to activate someone must speak a certain phrase. Once the phrase is spoken the agent self destructs within a few seconds. This killphrase is more of a security measure designed to eliminate traitorous agents rather than to eliminate evidence. One variant seen in the Human Revolution comics known as a Dragon-Fire implant kills the subject by releasing acid into the subject's bloodstream.
Nano-augmented agents have a similar killswitch. The main differences are that it doesn't blow up the agent and it takes around 24 hours to kill the agent, while older killswitches are practically instant. The way that the nano-augmented killswitch works is the nanites are programmed to go into exponentional growth.
It is subtly revealed throughout the story of Deus Ex that Paul and JC both have the Gray Death, and ultimately that their nano-augmentations ARE the Gray Death. As such it is clear that their killswitches amount to making them suffer the same symptoms as any victim of the Gray Death. A side effect of the killswitch is that non-lethal measures such as Gas Grenades and Tranquilizer Darts can kill JC rather than knocking him out.