I have a Nikon D60 camera. I know that I cannot use Yongnuo RF-603II transceivers to fire this camera remotely, but can they be used to trigger the remote flash with this camera?
Yes. The Yongnuo RF-603II triggers are manual-only. This means that they will work to fire any remote flash from any camera, as long as the camera uses an ISO-compatible hotshoe, and the flash uses an ISO-compatible foot: that is, the square arrangement with the rails and at least the central sync/fire signal connections: a pin on the flash foot, and a contact on the camera hotshoe. The rails are ground.
So, these triggers will work directly on any current digital camera hotshoe (some older Sony A-mount cameras may use the proprietary Sony/Minolta hotshoe, but the newer Sony models have iso-compatible hotshoes).
The RF-603II triggers come in specific Nikon and Canon flavors, but all that does is change the pattern of the pins on the bottom of the trigger to match the brand-specific layout for the wake-up signals. If you don't care about the wake-up function, you can use any of these triggers on any iso-compatible hotshoe--including those of mirrorless cameras. You just have to make sure that the on-camera unit is set to Tx mode explicitly, if it's not a Canon or Nikon camera.
Understand, however, that the only signal that's communicated is the one that fires the flash. Features like TTL, high-speed sync, setting power levels from the camera menu, etc. are not possible, as they require the other signals that are not communicated by these triggers. And the hotshoe on the top of the trigger does not do full TTL passthrough.