Some general statements:
Being skinny does not automatically mean that you are healthy
Being what people see as "very" skinny doesn't automatically mean that you're unhealthy
The BMI is notoriously full of crap
Your doctor is the only one who can tell you if you are healthy.
Weight is just one factor of health.
I honestly wouldn't obsess over it. Some people naturally store more fat or build more muscle than other people. Perhaps you don't store as much adipose fat as your peers. That's okay.
Just listen to your body. If you FEEL unhealthy then that's an indication that you want to start looking at your lifestyle and whatnot.
Barring that, just keep on keepin' on.