I'm not terribly familiar with the MechE program here, but I'll give it a shot.
BME isa very, very broad field and the undergrad BME program tries to train you with the basics to choose any one of the sub-specialties. The BME core curriculum includes basic biology, chemistry, physics, and math classes, a series of Engineering Physiology courses, a thermodynamics course, a programming course, a modeling/simulations course and an electrical engineering-type signals course. However, once you pick your concentration
, you take advanced courses in one specific area. My concentration is Cell and Tissue Engineering, so I take a lot of Chemical Engineering and Material Sciences courses. If I were to do a Sensors and Instrumentation concentration, I would take more CS and Electrical Engineering courses etc.
On the other hand, the MechE program is a traditional mechanical engineering program, and you will be trained as any mechanical engineer, but your upper level courses would focus in the biomechanics side of things. Another option to consider if you're interested in nanotechnology is the Chemical and Biomolecular Engineering program and the Material Sciences and Engineering program - it's really a matter of what you want your core education to be in. If you're sure you're interested in Nanotech, then ChemBE might be the best option. If you're interested in some application of engineering to medicine, but not really sure what, then BME fits the bill.
Hope that helps!