Sometimes a shop will refuse to replace only one side, just so they're balanced. But I would have expected them to say that about the rotors, not the calipers. Still, I'm not looking at your car firsthand...

When a caliper is sticking, most often it's the slide pins. Those can be removed, cleaned, greased, & put back together. Even if they're trashed, you can buy the slide pins themselves. Sticking piston within the caliper is less common, but it happens. If you've been driving for a while with dragging brakes, the high temperature could have damaged piston seals, etc.

Maybe they're right, but it doesn't hurt to get another shop to look.