Biomedical engineering or nursing

Yes you can.
A more normal path is to use biomedical engineering as the pre-med undergraduate degree for med school.
BME is a much tougher degree to do well in than is a typical pre-med program, and typically gives the med student a slight leg up in their med school entrance application against non-engineering pre-med applicants.

One might be able to make a higher salary in biomedical engineering.
Both are fairly technical fields, but biomedical engineering is even more hardcore technical than nursing.
Biomedical engineering will require math and physics that you’re not likely to need for nursing, even for advanced nursing degrees.

Bio-Medical engineering is the application of engineering principles to the fields of biology and health care.
Students can work with doctors, therapists and researchers to develop systems, equipment and devices in order to solve clinical problems.
Bio gadgets, bio medicals and RFID projects can be done.

Biomedical science is looking at advising doctors, from a medical science position using the diagnostic tests that can be carried out using technology in the laboratory and nursing is related to patient care to nvolving direct contact with the patient.