Should I train when I'm unwell?
No. However, depending on how unwell you are that would obviously influence your decision. You may even start to exercise and realise that it is much too ambitious for you and wisely decide to leave it.
On the other hand, you may get going and, thanks to the release of endorphins, feel a bit better... but later on that day or shortly after the session, the endorphins will wear off and the sickness will return, quite often worse than before.
Think of it like this; there is no gain to be made in training when you're unwell. Being unwell is a sign that the body needs to rest and recharge. To train is to deny the body of his healing function and asking it to spread what little energy is left too thin. If the body is throwing out symptoms of sickness and fatigue, don't use up it's precious remaining energy on a useless training session.
The body uses half (yes 50%!) of it's energy in digestion alone. This is why with sickness it is sometimes an idea to fast (as in stop eating!) to speed up recovery and allow the body to perform 100% on healing itself.
Feeling a little low emotionally is a different matter. Sometimes we can confuse this with feeling unwell. Feeling down is enough to stop an appetite for many things but in this case, exercise is obviously a great boost and healer and should be encouraged without excuse!