I interview and hire BMEs, so here is my take. BME programs seem to be maturing. You used to find programs that didn't seem to know what they were really about, but I see less and less of that every year. An example of a BioMed program I think is pretty good is Georgia Tech's. It is absolutely true that you have some flexibility in these programs, with some students doing more ME stuff, and some students doing more EE stuff. It is good advice to focus on what you find most interesting or would like to do as a career, most of the time you won't be able to get sufficient technical depth in both the ME side and the EE side of BioMed at the simultaneously.
Something that I would disagree with donpacino is that BioMed senior projects are not less complex than EE/ME projects in a well-run program. The complexity is simply in a different field, typically things like design control for medical devices, intellectual property, or the interaction between the design and the human body. You wouldn't believe how complex something with no moving parts can be when it is an implantable medical device.
donpacino's advice about internships is spot on. If you are interested in the medical field [which would be what a BioMed degree indicates], then relevant internships in the field are important both to expand your skills and network, and to help sell yourself to potential employers. A degree in BioMedical Engineering is becoming a good way to do what I do, but it is definitely not the only way to enter this field.
Having looked at that reddit thread, I think it is worth noting that medical device companies do hire BMEs with undergrad degrees, that you don't necessarily need a graduate degree to enter the medical device field, and that posting represents the experience of only a few. Masters degrees can be useful tools in the job market, but they are neither guarantees of your marketability, nor necessarily a good investment of time or money for everyone. Unfortunately, it is very, very hard to generalize here.