Randy and Julie McClure had three children who were long out of diapers and no plans for more when they heard about a program called Snowflakes, which arranges for women to become pregnant with embryos left over at fertility clinics. "We really felt like the Lord was calling us to try to give one of these embryos, these children, a chance to live," Ms. McClure said. Mr. McClure, though, disliked the fertility business, which he felt created extra embryos that were often destroyed or aborted. He feared that paying fees to receive the embryos would be helping an industry "that...