In my experience, additional basic science training did help in some of the preclinical part of medical school, depending on the subject. Undergrad material may, sometimes, in some ways, go into greater depth than medical school, but perhaps just in a different direction. For example, in an undergrad level immunology class we went into a lot of molecular level detail that wasn't covered in medical school. It can also give you terrible study habits compared to your peers if you coast along for certain parts. That was certainly my experience. However, there is a bit of insanity here of too much preparation, because the point is to go to medical school to learn medicine. That's the time you've allocated in your life to learn medicine, and that's when tons of resources are being allocated to you to help you learn. If you decide you want to go into a particular depth area later, you can do that. That's why people do things like fellowships and go to seminars and grand rounds and special tutorials at conferences and read on their own.
In theory you could go to medical school in another country and then come back and enroll in medical school in the US. Of course you'd know a lot of the stuff in advance, but you'd have wasted several years of your life so you can come back and review. Also, you wouldn't be impressing anybody with your knowledge once they knew your background; in fact, they might expect you to know even more. The same is true of taking time off to study a topic in general. If you take off 2 years and study immunology, you'll know the subject, and maybe the immunologists will be more willing to give you research projects to do, but unless immunology is your life's passion, it doesn't make much sense.
There are effectively infinite things to learn and you have a finite life.
It's more important to excel at something before medical school than it is to try to half way learn medicine in advance.
The one biology course that isn't a basic requirement which is very helpful is probably physiology. It will help a bit to put things together in your basic science curriculum (physics, chem, basic bio, o-chem) into the systems biology of a human being.
Overall though, I strongly support the idea of doing statistics or quantitative work in general. For some people, doing well in undergrad to get good grades to get into medical school is helpful. However, if you have some place in your course schedule, try to learn some of the things which are in demand which many people in medicine don't know well. That means you will have external skills to bring the table.
Computer science, economics, business, statistics, etc. are all things that are part of medicine and often aren't learned well by people before they go down the path of medicine.
In theory engineering can be really helpful, but in my experience what might be more important and capture what is useful is more along the design type hacking/project classes. If there are classes where you have projects and build things in some sort of fabrication lab, that is probably a better fit than a theory type engineering class for most MDs. Now, if you're interested in research, that's a bit of a different story. Also, all the good people I know in fields like electrophysiology seem to have a solid engineering background.
Right now, if you're looking to help a resident or an attending with the research project, probably the most valuable and desirable thing you can bring to the table is the ability to do statistics and/or computational work. Almost every project collects data (typically in electronic form), and they typically do not have the training to work with it.
It's important not to forget that medical school is a school, it's where you go to learn things. The point is not to try to know everything in advance. However, there is something to be gained by learning the important things which medical schools doesn't cover well, or things which can you can contribute to the field from outside.