Teach him Internet research skills...require him to present reliable, authoritative proof/statistics that documents that head injury is not a serious and likely complication of boxing. Reading the stats on injuries might convince him.
Don't feel bad about going back on your rule...if he'd said he wanted to spend his own money pursuing his hobby of smoking marijuana, you wouldn't have said he could, would you? Or if he wanted to buy a hot rod for street racing, or a gun to conduct armed robberies.
There are LOTS of times that parents cannot permit a child to pursue their interests.
Foster Cline, who wrote 'Parenting with Love and Logic' had a daughter who wanted to enter her horse in show jumping contests. He told her, 'If you are wearing all the standard safety gear, and are riding according to the instructions of your teacher, and you are injured, all the resources of the family will stand behind your rehabilitation. But if you refuse to wear a helmet, or you're riding recklessly, you might want to check out the type of care Medicaid will provide for you, because I will not take money away from your brothers' and sisters' college education, or your mother's and my retirement, to provide you with additional care. If you don't care to protect yourself, we won't protect you from the consequences of your choices.' Tough, yes...but his daughter always wore her helmet, he says.