No - songs don't need to be published with CCLI or any other organisation before uploading to Unified Praise. Publishing gives clarity over the authorship and intended use of a song, and publishing with CCLI will allow churches to report the use of your song, which in turn can generate revenue for you as the songwriter. Publishing your songs will therefore only be relevant for you if you intend to use them in a church worship context.
You can upload songs to Unified Praise whether they are published or not. Your song is protected by copyright, so you don't need to publish your song to ensure ownership.