I know of no factual basis for these statements whatsoever. I have heard them perpetuated many times at many places and consider them to be some sort of urban legend.
As indicated in Alexei's answer, there are people with meat allergies, but for these people, meat would be the "illness", not the transition to it.
The only reason for feeling unwell after eating meat for someone following vegetarian diet that I can see is a psychological one. I have experienced slight nausea and stomach pains after discovering that in the foods I ate, there were parts of animals, like their fat or bones/tendons (these are used in my country to give a certain "taste" to soups). It still does not match the violently part of "violently ill" though.
EDIT: As Azor-Ahai pointed out, in his case (vegetarian since he was born), the effects of eating meat can be physical as well, due to the body not being used to break down meat at all. I suspect this would be the case with long-term (10+ years) vegans/vegetarians as well. Do not take my word on it though.