How would I go about doing a PhD after an M.Eng in Biomedical Engineering?
Generally speaking, the main difference between an M.Eng and an M.S. is in research. An M.Eng is usually a course-based degree and usually does not involve any research. There is often some sort of "project" course which students will be required to take for credit, but that is still far from research.
An M.S would usually involve a research project and thesis, and it would give you exposure to the type of work a PhD student would be doing. That being said, many schools also offer course-based M.S. programs, so the name of the degree does not really mean much.

I would not recommend enrolling in an M.Eng program if you wish to study for PhD. In the U.S. in many schools having a master's degree is not a requirement for getting into a PhD program. I would advise you to inquire whether it's possible to be admitted to the PhD program at Cornell instead of the M.Eng.

For schools, the advantage of admitting students without master's degrees to PhD programs is that they will be around longer and probably have more time to do good work. In the case that the student doesn't quite meet up to their expectations, the can arrange for the student to leave with a M.S. halfway through the program. This to some extent compensates for the risk involved in admitting an undergrad (since they will likely not have much research experience).

M.S. students on the other hand have research experience. They have also written a thesis and possibly some papers. This gives the admission committee better insight into the applicants capabilities. Furthermore, a student with an M.S. degree will likely need less time to get started on their research and will not need to take as many courses, so they will probably hit the ground running once they enter the graduate program.

A student with an M.Eng degree will have a shorter PhD than the students coming in with B.S. degrees but will not have the experience the students with M.S. degrees have. So they will be at a disadvantage compared to both groups. Because of this admission to a "top 10" school would be a bit more difficult with an M.Eng degree.

Another thing to consider is that if your goal is a top 10 school, many of them give priority for PhD admission to students already pursuing M.S. degrees at their school. These are typically the schools which require master's degrees for PhD admission (such as Stanford and MIT).