Jeremiah Clayton Jones discovered that his former fiancée was pregnant just three weeks before the baby was due, when an adoption-agency lawyer called and asked if he would consent to have his baby adopted. "I said absolutely not," said Mr. Jones, a 23-year-old Arizona man who met his ex-fiancée at Pensacola Christian College in Florida. "It was an awkward moment, hearing for the first time that I would be a father, and then right away being told, 'We want to take your kid away.' But I knew that if I was having a baby, I wanted that baby." Mr. Jones...