This won’t be the first time a politician has offered up an anecdote to support their position that was later proven to be fallacious.
It generally tends to be a Democrat though.
However, although side-effects of vaccines are often rare, they can be devastating to those who happen to be susceptible. We have a recent event in Scandinavia, where it turned out that the vaccine against swine-flu caused narcolepsy in sensitive subjects. According to publications in medical literature, there are indications that also the HPV-vaccine can cause severe neurological side-effects (though, the cause and effect is not yet proven).
Demyelinating disease and vaccination of the human papillomavirus
Human papillomavirus vaccine safety in Australia: experience to date and issues for surveillance.
Although, one talks about sensitive or susceptible individuals, it is often the case that one cannot know prior to the event who is sensitive or not. That has of course a very strong bearing on the question whether to make vaccinations compulsory or not.
PS: The above was meant as a general point re vaccinations. I know that there was an opt-out clause in the (not approved) Texas legislation.