In which format are the character skeleton rig and the animations provided? In general I'm afraid there is no automated way to transfer rigged and animated sprites from Unity to Spine yet.

After having taken a look at the WebGL demo, I would not want to transfer the included animation and rig anyway .

The best way here would be to create a new character rig in Spine, add all images to the Spine project and create skins for them. If you really want to switch every single attachment separately, you need to write the respective code that changes Spine attachments instead of Unity's sprite images.

Please note that the spine-unity runtime comes with some example scenes such as Spine Examples/Other Examples/Mix and Match Equip which demonstrate how you can change attachments.

You can also use Unity Sprites as attachment, as described shortly on this forum thread:
Attaching animated sprite to spine-skeleton
This requires a base Spine Skeleton to switch out the respective attachments as well though.