Skin can become cold and clammy due to a variety of reasons including emotional responses such as fear, too little fluid, or even a drop in blood pressure.
Being sick with the flu or any other sort of infection can cause your skin to change and you may feel hot within yourself, but be cold and clammy to the touch. This is caused by your body working hard trying to fight off the infection.
Poor circulation may be responsible for this, as well as conditions like shock. With poor circulation though, there may be swelling in extremeties and discoloration of the skin as well.
Illness can cause symptoms of skin. The body will try to fight infection and the internal thermometer is compromised. When the bug is gone the body should go back to normal.