If what you describe as a "vernier" is the older all-metal style and not the newer version containing electronics (properly termed digital calipers), then a coil-type demagnetizer will be suitable. By coil-type I mean the style made from a transformer, commercial demagnetizer as used in shops with magnetic chucks, or a bulk tape eraser from the electronic antiques store. If the calipers contain electronics, then I would not use any of the aforementioned. They might survive or you might create an expensive paper weight.

Steel wool is a real problem around magnetized tools. If you use it, make a bench sweeper to clean the bench. A magnet in a plastic bag will gather up the fines. When finished sweeping, hold bag over trash can and carefully remove magnet allowing fines to fall away into trash.