"We had a son on Dec. 26, and he was born with multiple issues that we confronted and had to deal with as we were moving through the last couple of months at Mass General," Hill said. "Unfortunately he succumbed and he has passed. He taught us a lot of things, and unfortunately things didn't work out.
"My wife has been extremely strong. Our son, who is 2½, is with us here down in Florida, so we are going to enjoy our time here and obviously make the most of the opportunity that's here to present itself to play baseball."
Hill, a Milton, Mass., native who pitched for Boston from 2010 to 2012 and made 63 appearances with the Cleveland Indians in 2013, signed a minor league contract with the Red Sox during the offseason so he could remain close to his family.
He expressed gratitude toward the staff at Massachusetts General Hospital and said the Red Sox "have been tremendous."
Hill said he is glad to be back around his teammates and baseball after such a devastating period in his life.
"Excited to be here, to be playing baseball again and get back into the normalcy of my profession and to be around a great group of guys and to just take that next step," he said.
Hill also said he is ready to focus on pitching again.
"I fully feel that the opportunity is here," he said. "I think that was also with a blend of the opportunity that is here and obviously with our family situation that we had. With the opportunity that is here, [I want] to make the most of it and do everything I can on a daily basis to perform."