No, according to all the baby name sites I've seen (including the one on here), it says that
Regan means little king for a boy and regal and royal for a girl.
Ronald Reagan had an "a" in it. I was just kidding about the Shakespeare thing. I said it because
Regan was one of the bratty daughters. I wish she was the good daughter. Then I'd be thrilled to say I got the name from Shakespeare.