Having such a negative reaction is such a gross anomaly that basing your health around it is like refusing to eat anything because one guy you know got struck by lightning once while buying food.
A better analogy would be refusing to eat peanuts because a small fraction of population is strongly allergic (anaphylactic shock, as opposed to hives) to them.
However, discounting reactions to vaccines out of hand is also sticking a head in the sand. As The Little Raven's story suggests, vaccines do carry some risk.
Influenza vaccines (and some, but not all others) are grown using eggs as a medium for viral reproduction. In the case of the little girl, my guess would be she had an violent allergic reaction to a component in the vaccine; possibly the eggs. I know that I have have a strong allergy to eggs myself, and because of that I can't have any egg-grown vaccines (and had a medical waiver for the MMR requirement). Injecting any egg product directly into the bloodstream would most likely be Very Bad (tm), seeing as how bad a reaction I have simply ingesting them.
(Typed out and deleted some science-y speculation as to safety measures)
There are numerous ethical issues surrounding vaccination, and both sides have arguments that have merit. I'm not talking bunk like "thimerisol causes autism," which has been shown time and again in scientific studies to have no correlation.
A good example might be the need for herd immunity versus individual freedom. In order to fully protect a population from a virus via vaccination, a certain percentage of people need to be protected. This percentage is typically in the 70-80% range. However, at least in the US, people have individual rights that include choosing whether or not to be vaccinated. As can be seen with the various flu vaccines, that is not a number that typically gets reached naturally. Something like influenza isn't typically associated with a terrible effect on the populace. However, if you take something like polio (and ask any octogenarian about polio scares) there's a clear need to have that herd immunity. Hence, the requirement for vaccination to attend grade school. That tends to be a lot less controversial, since it had much worse effects. At what point is that line where protecting a population trumps individual freedoms? That's not an easy ethical question to answer. In fact, I'm not sure it's even possible to answer it. A great example of that ethical dilemma is the push to mandate the HPV vaccine.
Of course, that's not the only problem here. Polio is pretty much removed from most "industrialized" populations. There's a clear effect of vaccination. However, influenza's rapid mutation rate means that no vaccine will ever be completely effective (and natural selection most likely won't be either). That begs the question, should we even try? Well, ask anyone that's lost a family member to influenza (of any type) and the answer is yes. Yet, that mortality rate is relatively small. Again, the question is, where do you draw the line? In this case, is the combination of scientific effort and monetary expenditure worth the knowledge of virology and protection from infection that the flu shot gives? Some people will say yes, some will say no, and the truth is there is no clear cut answer.
So, to sum up, to belittle someone's beliefs on a complex ethical issue is betraying an ignorance of ethical complexities that present themselves. There are issues that are clearly borne out of willful ignorance (ie, thimerisol causes autism). However, there are a host of issues that are borne out of ethical dilemmas. While you may not agree with the conclusion that someone else has made as a result of attempting to reconcile those dilemmas, that doesn't make their conclusion invalid nor incorrect. It means that they have a different opinion that you.