Not an answer to your question, but a possible option to consider when there is a delay caused by a mechanical issue, weather delay, or medical emergency - request a "fly over, fly back". In your case, you would fly to DEN and because the mechanical delay caused you to miss your connections, request to be flown back to BOS from DEN. You are entitled to receive all of the miles booked on the original itinerary. One must wait 5 days after the date of travel to contact United to request the miles.