I've got a BS in CivE, and I'm in the class of 2013. I didn't originally pick CivE as a premed major. . . long story, but came back to medicine a couple years in.
I see why you are saying this, but it is a gross overstatement. I ended up doing research in a civil engineering lab that was concerned with the inactivation of airborne pathogens, which certainly doesn't qualify has having "...absolutely no overlap. . ." with biomedical science. I also was able to take a lot of medically related classes through engineering that counted as upper division classes for my major, like environmental microbiology, and environmental health for developing communities (taught by an MD).
That said, there are a lot of classes you have to take in civE that you will be really frustrated with if you are actually interested in medicine. My transportation engineering class my senior year was a total disaster, and a lot of the construction classes are a total drag.
Don't let the med school prereqs get in your way. If you are smart, you will plan your schedule so your premed requirements fit as electives for your major. At my school, for instance, you could count OChem as an upper division elective. Just make a case for why the class should be counted, I think the department will be reasonable. Even with a reduced time frame to tackle the premed requirements, I fit everything in and only had to delay my graduation until the summer.
Now, I have to say that I am now getting a masters in biomedical engineering. And if I had it to do again, I would look a lot harder at chemE (not with a bio double, though, we're getting into marginal returns now). Although I don't think you would get quite as strong of a mechanics and structures background with that.
If you really like civil engineering because you want to know how the ventilation in an OR is special, or how the hospital stands up or is constructed or something, do it. An engineering degree gets respect.