Can my 16 year old legally refuse to attend boot camp?

Most likely not. A lot of boot camps, for troubled, teens only accept individuals that have been sent to them by a judge's orders. There are private boot camps that allow parents to pay to admit their children. But, when you are dealing with a 16 year old who is old enough to have the right to drive, work, drop out of school and gain emancipation from their parents, I doubt that you could force them to go to boot camp without a judge's orders.