Should an advisor care if their student becomes overweight? So your student is becoming overweight: does this negatively impact your research group in some way? Should an advisor care?
Becoming overweight is not an issue per se.  Major physical changes, however, are often a sign of an major ongoing mental or physical health event.  Rapid weight gain, for example, is often associated with depression or thyroid problems, both of which can have a major impact on the student's ability to learn and work.
As such, I would definitely be concerned if a student was visibly gaining a large amount of weight---not because of attractiveness or disapproval of weight, but because of what else it might indicate.

I would find it highly inappropriate, however, to bring this up with the student by saying something like "I've noticed you've gained a lot of weight" because there are also a lot of more benign reasons somebody might gain weight, including recovering from mental or physical health problems that would also be none of my business.  

I would, however, notice it as a possible red flag, and start to keep a more careful eye out for other signs of distress, which might legitimately trigger a conversation about, for example, mental health.