I know a woman who was a victim of domestic violence. Her abusive ex beat the sh%t out of her one day and tore her face up pretty good. She has a really nasty scar on her eyebrow that hair cannot grow out of. The scar is cleverly hidden by a tattoo that looks like normal human eyebrow hair. It is so good that, unless you know to look for it, you would never notice it.
I would hope that, if a UPS employee were in a similar situation, some sort of hardship exemption to the tattoo rule could be made.