Read and understood that your boyfriend has vomiting, diarrhea, burping extremely strong rotten egg odors and light headedness.
All these symptoms are suggestive of gastroenteritis and he need a proper clinical evaluation for dehydration, septicemia and control of vomiting.
I would advise you to take him to the hospital/ER so that proper clinical evaluation will be done as well as he gets a prescription for the medicine like PPI, Domperidone, an antibiotic, probiotic and supportive treatment.
Burping of the strong rotten egg odor is due to bacterial overgrowth causing / producing gases giving this typical smell.
Hope this helps.
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