No, children are not welcome in the spa for a haircut. Get them a cheap cut before going on board.
On the one hand, you'd think they'd want the $ from cutting the kid's hair. On the other hand, spa appointments usually fill up, so a kid appointment would be one less adult they served. Some people go to the spa to get away from kids (I know, if you don't want to see kids, why cruise Disney, but....) Some kids handle things well, others squirm, cry, fuss, etc. So...the answer is NO kids in the spa area except for the time when they run the teen special.
Kid hair braiding is available on deck by the pool.