My mum had a talk with me about that too when I said I was going vegetarian. My BMI is exactly the same at the moment (But I'm a shorty! 63.5 and 118-119 pounds), I was a chubby kid as well (as in, clinically overweight by 1 pound) and I think my family is still not used to a normal me. I think it makes my mum feel kinda bad too, because she is overweight (well, my whole family is except maybe my little brother, since he is still growing), but she has a thyroid condition and 3 kids, so it's hard for her.
Just show them that you are as concerned for your health as they are, but you are also concerned about the ethical implications of eating/using meat and animal by-products. Try to provide them with research about how healthy a balanced vegan diet can be.