The real answer is that we (scientists) don't fully understand how this happens yet. But, the current thinking on it, and that is supported by the studies that have been done so far, is that those feelings you described are all psychological stressors. When someone experiences them, their body's stress response systems are activated. These include things like an increase in cortisol, a hormone produced by the adrenal gland, increased heart rate, etc. Basically, many of the same systems that would be activated for the classic "flight or fight" response. However, rather than the perceived danger coming, and quickly being taken care of, either by fighting an attacker or running away, with these other psychological stressors, "fight or flight" doesn't work and the stress sticks around chronically (long term). Some of the signals for stress remain, but the body doesn't respond in the same way when they stick around for a long time. The best example is that of cortisol. Even though the stress stays, cortisol concentrations decrease again in long-term stress and don't stay elevated. It just occurred to me that I don't know what happens to cortisol when a chronic stressor is finally removed (or goes away). [If anyone knows of a study where they've followed cortisol, or corticosterone (the rodent version of the hormone), concentrations in an animal after a chronic stressor was removed, please post a reference. I'd be curious to see it.]
Anyway, the main focus at this point has been on the interaction of cortisol, with components of the immune system. That's the point at which my knowledge ends on this. I know from the endocrinology perspective that such studies exist, but I'm not sure what detail they've reached on the immunology end of things yet.
The reason I'm curious about what happens to cortisol when the stressor is removed is that on the occassions when I've been under a lot of stress (important deadlines with a lot of work to get done for them), it seems I get sick right after the stress has passed, not while the stress is still present. Of course, that's on the immune side of things, and there are other health problems that can result from chronic stress, such as high blood pressure, heart attacks, stomach upset/aggravation of ulcers that happen during the stress. So, I guess it also depends on what you meant by sickness.
This is a growing field of research. Here is the website for the Psychoneuroimmunology Research Society, which includes links to their journal
Brain, Behavior and Immunity that focuses on the type of research questions that are aimed at answering your question.
http://www.pnirs.org/
