I believe the "veil" is actually the amniotic membrane, also known in folklore as a "caul." The beginning of "David Copperfield" states that David was born with a caul, and it was thought to mean the person would be protected from drowning. Here is some more information from a site devoted to midwifery. I can't vouch for its accuracy.

Vance Randolph wrote a book called "Ozark Magic and Folklore" (formerly published as "Ozark Superstitions"), which I read long ago. I can't remember whether it covers the specific topics you ask about, but it contains some fascinating information.