They never go on a vacation during the winter? Did the house freeze up while they were gone for two weeks? Does their furnace have a habit of dieing?
When still working we'd take two two week trips a winter to Mexico. Now that we're retired, we've spent most of the last two winters outside. Never had any problems.
I wouldn't winterize the house - too much hassel. There are other simpler and less expensive things to do. There are devices that will call your cell or someone else's phone if the temp in the house drops below a certain point. There are lights with temp sensors that can be set up to shine at the neighbor's house if the temp goes too low. Have a neighbor check the house. Hire a security monitoring company. After the initial equipment install, the house is monitored for flooding, temperature, break-ins, etc for $25 per month.
It sounds like they either don't want to leave and are looking for excuses, or they just worry too much.
Bill