I'm not a current employee of UPS, nor have I worked for the airline part of the operation. I have, however, worked in the ground operation and my answer is based on that experience.
There are Saturday package deliveries at UPS that the customer has to pay extra for. Other than that, the ground operation is a Monday-Friday operation. The ground hubs are closed on Sundays, and for the most part on Saturdays too. There may be some limited big-truck operations over the weekend, but I would even doubt that as most of the trucks seem to be day-cab shorter haul routes (Nashville - Knoxville - Nashville, etc.).
I would venture a guess that the airline operates 24/7, but this spokesperson was probably referring to roadway traffic and the difficulties that snow and ice can cause.
Tim