Yes. As long as they prohibit ALL redheads, not just enlisted or just women redheads...
He'd probably be better off propping the decision up with some carefully selected findings. Maybe show that redheads are more likely to be mistreated among serving members, with a .03% greater chance of suicide if the DOD doesn't pay for their hair coloring. Kicking them out is simply protecting them. Maybe not all of them, but we can't ignore the .03% that would be at risk.
Science cannot dictate unit cohesion. If enough sailors/soldiers are convinced that they'd be uncomfortable around a transgender member, they'll be uncomfortable around a transgendered member.
The funny thing is, it sounds a LOT like the comments about putting women in combat or on deployed vessels. Or allowing gays.
I suspect people said the same sort of thing about 'what if a white soldier has to take orders from a black officer?' or 'How will they handle the bathroom facilities on a seagoing command?'