I've been looking into doing the five year MD/MS in biomedical engineering and the requirement at my school is a bacc. degree in engineering or a physical or biological science. If you don't have a degree in engineering, the additional pre-requisites are linear algebra, diff eq and physics, but I don't know anything more specific. I'd say check out if your school has the 1 year masters available, and if not, look into other masters programs you might want to do after med school and their specific requirements. I very much doubt an engineering PhD program would take you directly, but if you do a masters in engineering with thesis option, it will be much more accessible.
I did my undergrad in engineering, and I can say a masters is sufficient for many engineering jobs, but especially the biomedical industry has a preference toward doctorate level. Not sure how the MD plays in there and whether it substitutes for a PhD.