You can run the furnace if you like, though I'm not sure I would do it in a trailer (because it would be left unattended while you drive).
The fridge itself won't freeze and neither will the contents if you start out the day at normal fridge temperatures. The insulation keeps both heat & cold out.
Water lines may be a different story, but if they are not frozen to start with they should be ok for a day of travel.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL