TeenHealthFX cannot tell you if you are skinny due to not eating meat. We do think that it is possible to be vegetarian and be healthy. The trick is to eat a well-balanced vegetarian diet. Check out our answer to question Vegetarian - How To Eat To Stay Healthy for tips on how to achieve this.
Feeling depressed because you feel so skinny is something that TeenHealthFX recommends you address this issue with your doctor. He or she can evaluate your health and make recommendations for you. You could also benefit from meeting with a nutritionist who could help you implement a healthy vegetarian diet that would satisfy both you and your mother.
Again, TeenHealthFX recommends that you speak with your doctor about feeling so skinny and eating a vegetarian diet. If you do not have a doctor and live in northern New Jersey, you can contact the Adolescent/Young Adult Center for Health at 973.971.6475 for an appointment. Outside of this area contact your local teen health clinic.