I tried to get the sample project running, but to no avail. Your project setup is not great, all Spine runtime files are just dumped in their, making updating them hard. Here's what I did:
Remove Source/Spine in Xcode, using "Move to Trash"
Remove Source/spine-c in Xcode, using "Move to Trash"
Copy the contents of spine-c/src/spine to Source/spine-c
Copy the contents of spine-c/include/spine to Source/spine-c/spine
Copy the contents of spine-cocos2d-objc/src/spine to Source/spine-cocos2d-objc
You can find the latest spine-c folder here, the latest spine-cocos2d-objc folder here.
This will give you an up-to-date version of spine-c and spine-cocos2d-objc. Once I did that, compiling still failed as the support library from SpriteBuilder is not present, so CCBReader and CCShader can't be resolved. Everything else appears to compile though.
Sorry I can't be of more help, I've already spend 45 minutes on this. If your problem persists, please modify the spine-cocos2d-objc example to reproduce your issue. Otherwise I'm not sure what else I can do on my end.