As a funeral home owner and operator in Spartanburg, South Carolina, the Rev. Lawrence Meadows knows what it means when doctors start talking about hospice.

“That implies you only have about 14 days to live,” he says. “And the next thing you know, you’re getting a sponge of water rubbed across your lips and maybe some ice chips.”

That prognosis was unacceptable to Lawrence, who was only 39 last October when he was diagnosed...