How can I play the songs in my iPhone library that has a square with a circle around it

That usually indicates an incomplete download. Did you try tapping the song to see if it would continue to download and play? Also be sure you are either connected to wifi, or have Use Cellular Data turned on in Settings>iTunes & App Store so you can access it over your cellular data connection.

Actually I found if I set Sharing (Music>...>Sharing) to my iMac's iTunes then they all downloaded. Where this choice didn't exist on my iPad, I just deleted all the songs and synced then added the right playlists and songs back and they all came back as they should.