Sorry to hear about your plight.
From what I understand recovering from burnout takes time. Are you prepared to put other things aside and focus on getting well again?
There is an account of how Richard Feynman dealt with his burnout that I find is well worth reading. It's available in the later part of the chapter "The Dignified Professor" of "Surely You're Joking, Mr. Feynman". Excerpt here:
http://just.thinkofit.com/feynman-on-work-and-play/

While disappointing in a way, the advice to eat and sleep well really is important, and it would be easy to underestimate to what extent.

I have it anecdotally from a very competent physiologist in the field of sleep research that when they were studying burnout they have found no exceptions to disturbed sleep in the sufferers. Sleeping well is about more than just getting your hours in. The quality of sleep can be very low without the person subjectively noticing micro-awakenings and disproportionate time spent in shallow sleep phases. If your conditions permit I would suggest getting in touch with a sleep laboratory and request their help to evaluate the quality of your sleep and if necessary their help to improve it. In many cases these would in any case be people with much knowledge about stress and hormonal systems; well worth talking to. If you can't get access to such a place, maybe you could get an indication from DIY approach.

About the food: I don't expect this to be a popular opinion with you or with much of the community in the Valley, but I'm of the opinion that a vegetarian diet, unless incredibly well conducted, does not help, but is detrimental.
I say this as someone who is a tentative proponent for "evolutionary nutrition"/"paleolithic diet" or whatever you may want to call it. My suggestion would be reading up on that and trying it for a few weeks. At a glance it would involve dropping intake of grains, dairy products, refined carbohydrates (especially sugar), legumes, and keeping salt íntake much lower than most westerners. Switching diets, especially from vegetarianism is a step most people would not accept, so my respect if you try it, and you have my understanding if not.

It's becoming increasingly clear that states like depression have a strong connection to inflammation. That's relevant for both the food and sleep components, but it may also be worth trying to suppress inflammation a bit and see if that helps. Preferably get medical professionals help setting the regime, but an easy one would be trying paracetamol with ibuprofen or acetylsalicylic acid for a week or two (check their instructions, I can't remember the upper time limit for continuous intake without special measures)

If you have a hard time with sunlight, I would advice trying to change that. At the very least, make sure your vitamin D levels are sufficient. You may wish to consider supplementing.
Strong light also helps setting and maintaining the circadian rhythm, which is of importance in this context.

For many people, physical touch may help. If it suits you, seek out activities that promote that.

You mention you spend much time thinking of negative past events, and I think that may be an important and insightful observation. If you're not already aware of it, please look into the research on psychological time perspective pioneered by Philip Zimbardo and John Boyd ( see e.g. http://www.thetimeparadox.com/ , http://fora.tv/2008/11/12/Philip_Zimbardo_The_Time_Paradox). The interventions they suggest to change the proportions of use of time perspectives might be helpful.

Not directly related to you symptoms, I might add that I think there should be no end to the "learning phase", so if you're doing it right, that's the only time possible to get burned out. There is no deadline. You never need to stop learning.

I hope you recover well. Good luck!