The song was written in 1780 and used the term "Colly Birds" which is, and was then, an arcane reference to blackbirds, perhaps ravens. The song has been rewritten many times since the and has changed the term, but it wasn't until 1909 that the term "calling birds" was referenced. The ecumenical church, seeking a reason to sing the song in a religious setting, decided the "four calling birds" were the Gospel writers Matthew, Mark, Luke and John carrying out "the Great Commission", that is, to spread the Gospel (it means "good news") to all the world. In a word, they were evangelists. In the second Charlie Brown cartoon called "It's Christmastime Charlie Brown", Linus Van Pelt tries to explain to Sally Brown the meaning of the phrase and relates it to 1 Samuel 26:20 in the Bible. In that story, David, before he was king, was being hunted by Saul, the current king, to be killed. Needless to say, he was a little concerned about that fact and, as a metaphor, said the hunt was like a partridge being hunted in the mountains. So, Charles Schultz used Linus to explain it like so: Linus: A "calling bird" is a kind of partridge. In 1 Samuel 26:20, it says, "For the King of Israel has come out to seek my life, just as though he were hunting the calling bird." There's a play on words here, you see. David was standing on a mountain calling, and he compared himself to a partridge being hunted. Isn't that fascinating?
Bottom line, it has morphed from when written. Ornithologically speaking, there is no such thing as a "calling bird". Since it was a gift, you'd assume it was a melodious sound, and thus, feel free to ascribe any black songbird you care to imagine. I'll pick a Black Redstart just due to it's rarity, but it seems likely it would be a Crow or a Raven.