I hope I can shed some light, as this is almost exactly the path I took. I majored in molecular/cell biology and wrote my undergrad thesis on plant biochemistry and genetics, and (after a several year gap, during part of which I worked in a government research lab) recently finished up an MS in mechanical engineering.
@WayOutWestMom is correct, generally speaking. Requirements will vary from program to program, but most programs will require you to have taken a number of prerequisite courses before applying or will require you to make them up after being admitted. Some programs are more strict about this than others; some will not even consider students without an engineering Bachelor’s, but most will. I was admitted to several well-ranked MSME programs despite having no real engineering coursework/experience. One mitigating factor might have been the fact that I applied for research-based programs and had a solid understanding of the research process, and my background was a decent fit for the research areas of certain faculty members.
All that said, as a general rule, you should expect to complete a number of engineering pre-reqs before applying. Even though I didn’t, I wish I had–it would have made the transition much, much, much, much easier and smoother.
Regarding your other questions: yes, it would be highly advantageous to pursue any engineering degree, including BME/BioE, regardless of your undergraduate degree. This comes with a huge caveat, though. The engineering MS without an engineering BS may, at least in the eyes of employers, make you less qualified for many jobs than someone with just an engineering BS, but it may make you more qualified for many other jobs, too. It may also make you less appealing a prospect than someone with both a BS and an MS in engineering, at least when you’re starting out and don’t have much experience yet. And this brings me to the second part of the caveat: in the engineering industry, experience is king. Having the engineering MS without any experience can make it extremely difficult for you to get a job, even an entry-level job. Consequently, if you do end up going for an engineering MS–especially in BME/BioE, a field where not having experience essentially guarantees you won’t find a job–MAKE SURE YOU DO INTERNSHIPS AND CO-OPS. I wish someone had told me this during my MS. If you don’t do any internships or co-ops, you may well be wasting your time getting the engineering MS after a non-engineering BS. So let me repeat it again: DO INTERNSHIPS AND CO-OPS.
Good luck, and let me know if you have any more questions.