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You don't need to host your game on clay.io to use their leaderboard system (crazy & over-generous if you ask me!).
I don't know how to use the leaderboard plugin with cocoonjs although it does appear to be supported in the documentation - and I have no idea if it would work with phonegap (might give that option a try myself in a while....) or intel xdk.
The clay.io plugin is rich with features and with some simple planning you can display the leaderboard embedded in your game using text objects without having to resort to using the clay.io popup window.