I guess it depends on who said that and what they meant by it. This is how I interpret your directive:
On one hand, for a med school app, you probably don't want to get on a soap box and go over anything that is controversial. I think the consensus is, for the most part, keep away from things that could potentially cause anyone to disagree with your point of view. If you feel like you have to cover something controversial, ride that fence, but err on the side of bleeding heart. In that sense, try not to make a statement. If someone on adcom disagrees with you, you're not making it easy on yourself. For the most part, pre-med and early med students don't have the information they need to really make a real statement.
I feel like the way to go on med school applications is to simply "tell your story." It was obviously very personal, but I really don't feel like I made a statement. I just tried to say why I wanted to pursue medicine, why I was different than everyone else, why I would be a good choice, and how I knew it would fit me (experiences). Not that what I did was necessarily correct, but I got good feedback from my proofreaders and got accepted.
I guess it all depends on your definition of "statement." My definition is declaring where your beliefs are on an issue. I think it's a bad idea (at least on the AMCAS personal statement) going into med school.
As for your second question: maybe. At this point (interviewing for residencies) in a career, you should have a much better idea of what is important to you and what you want to accomplish. Residencies are also interviewing you to see if you would be a good match for their programs. A show of confidence would probably be better received in this situation that is largely based on competency, and a researched, tailored statement could also align your goals as a physician to a hospital's goals for healthcare. I've not had this experience yet and could be way off base, but that's my take on it.
Interesting questions.
