"I research my health problems on the Internet. Am I a hypochondriac?"
First, we should ban that word when talking about ourselves. No one wants to be called that, and doctors who use that word are committing malpractice. Everyone has some range of complaints and worries in life, often physical and mental together, and this is our job as doctors: to hear them out. I firmly belief that no complaint is illegitimate.
Nor, for that matter, is looking stuff up on the Internet a problem. Given that the advice proferred by doctors is most often not consonant with the scientific evidence (to quote this much-cited paper, "Even those physicians who are most enthusiastic about EBM rely more on traditional information sources than EBM-related sources "), I doubt that seeking information on the Internet is any worse. (Of course, there is already a considerable literature on the topic.) Nor do I know of evidence that seeking health information on the Internet increases worry, another common concern mentioned to me.
I think the relationship between physical symptoms and worry actually go in another direction, at least as I see it. A subset of those who have common physical complaints sometimes have a lot of them, and with some frequency these are due to undiagnosed anxiety disorders or other psychiatric ailments - which often go untreated.
In short: physical complaints are never to be dismissed, no matter what their cause, and having recourse to publicly available information is not to be looked down on either.