DON'T DO IT.
(By the way, nice user alias - I volunteered with a gamma knife unit once before, and thought it was one of the cooler clinical volunteering experiences I had. It's also the only place where I encountered a real, specialized physicist outside of an academic lab or classroom.)
I was an environmental engineering major for the first half of my undergrad (it's closely related to civil engineering). This will vary school by school, but for us it was admitted that the engineering program was a five-year program that was crammed into four years. They accomplished this by making nearly all of our engineering classes two and three units. In theory, one unit is supposed to equal some number of hours required for a class. Those classes were easily worth four to five units on a time commitment scale, yet the paltry number of units meant that you were putting in a ton of work for something that wouldn't raise your GPA by much.
For basic sciences, I was required to take four semesters of math classes (ugh, overkill for medical school), four semesters of general chemistry (two gen chem, two ochem), two semesters of physics, biochemistry, and molecular biology (but not the first two semesters of general biology - these classes are "recommended" but not "required" for medical school).
The engineering courses varied in how interesting they are. Courses like statics and fluid dynamics were all math. There were some chapters where the textbook's explanations had more equations and numbers than they did letter-based sentences. Courses like wastewater management were interesting (perhaps partly because of their real-world application), but still involved a lot of math while drawing on concepts from chemistry and physics. Engineering microbiology was a course that I enjoyed, because it was a lot like microbiology (another course I liked), only more applied.
All right, none of that directly answers your question. I'd sum it up like this: you will need to really like math to not burn out of engineering (even biomedical engineering). You'll invariably have to take some classes for your major that don't directly interest you, whether because they're very academic (seemingly little real-world application) or because the subject seems unrelated to your major and interests. When you do, will all of that applied math bore you into a bad grade?
You'll also need to really like science, because many of the engineering classes draw on scientific concepts from other classes. That puts more of a strain on you compared to the standard pre-med who takes a class, aces the memorization-based exams, and then promptly forgets 99% of the material. Speaking of memorization, the approach to a number of my engineering classes was rather different from the standard pre-med classes. Rather than memorizing, you were taught for understanding. There were no equations to memorize, because you were expected to derive your own on many exams. It's very satisfying to be able to do that, but it's a different learning style and expectation.
Another interesting bit of info: engineering classes do not count toward your science GPA. For what ever reason, it's not considered a science degree by AMCAS or by the medical schools I've encountered.
Perhaps the biggest reason to avoid engineering (unless you want to go into engineering) is this: despite the fact that it's widely known that engineering is one of the most difficult courses of study you can achieve, admissions committees don't care. Hell, back when I was doing it, the national average GPA for engineering majors was 2.8. Despite that, I have not encountered a single medical admissions person who cares about that. I've encountered plenty who were willing to say that a candidate with a 3.3 and an engineering degree was less desirable than, say, a history major with a 3.7 (no offense to history majors). In other words, even if you do an amazing job and get a high GPA, you'll be regarded as equal to those who took an easier path. Self-satisfaction is worth something, but the payoff here isn't worth the risk, in my opinion.
I ultimately realized that I didn't think in math well enough to succeed, and changed majors in an effort to salvage my GPA. My semester GPAs shot up, but it didn't pull up my main GPA enough to make me competitive. I had to do a graduate program, and even then I struggled to get into medical school (although I made it). I'm angered by the fact that committees only look at the GPA as a number, and seemingly don't weigh other factors like the type of classes taken or how many were taken per semester. It strikes me as being very unfair to those of us who willingly pursued a difficult course of study, when we could have just as easily gone for a pushover major.
The advice against majoring in biology comes from the fact that it's a bit harder to distinguish yourself from other pre-meds, and that the current trend that's been running for a few years now is for medical schools to choose candidates from a non-science background. You can still make it in with a degree in biology. Chemistry is a fairly uncommon degree, from what I've seen.
If medical school is your goal, make life easy on yourself: major in psychology. It's a very interesting topic that can have parallels to patient care, and in my experience, the classes are easy (helped in part because they're so interesting). I've encountered quite a few medical students who majored in psychology, but I don't think it's become common enough to cause a backlash against it.
Take it all with a grain of salt, but I wrote up this huge essay because I feel strongly about this. I and others like me are wronged by the admissions process, I feel, because the system penalizes us for any poor performance without crediting us for the harder work load that we carry. I hate to see others going through the same thing that I did. It's not impossible to excel in engineering, but it's not a feat that's easy for most people. If you do choose this, I sincerely wish you the best of luck.
