Pregnant daughter moving out-of-state, must she notify baby's dad in wrinting or orally to be in compliance?

As long as the baby isn't born yet, she can move wherever she wants and she doesn't have to give the boyfriend any kind of notice.  Indeed, even after the baby is born, if there's no wedding and no court order, the birth certificate is all the evidence she needs of her sole legal custody.

However, moving before the birth is a good idea, because once the baby is born, wherever that is becomes the baby's "home state," and a custody application would have to be filed in that state alone, until and unless the child lives in another state for at least six months.