I can understand your employer wanting nothing other than traditional piercings (ears, usually no more than 1 pair and less than 1" and no large hoops). But hair length shouldn't be an issue as long as it's out of your face and tied back since you work in a dental office because it could be a hygiene issue with loose hairs coming off and getting on people or on the equipment.
I will wear my hair down but I keep my bangs out of my eyes and I wear headbands to keep my hair from falling forward. I work in a very casual business office. I think my hair is fine down and long provided it's clean and groomed.