There's a substantial amount of evidence that your name can actually have an impact on your life choices, so names clearly aren't just arbitrary. In this case, it may be hard to trace any definitive connections, but let us make the effort. If the name makes a difference, it could be either based on the sounds or the connotations --or the connotations of the sounds. So let's analyze those:
There may be no "Hey John," but as a counterexample, we can take The Premiere's "Farmer John". Could the title characters be interchanged?
The sound: "Joe" and "John" are both one syllable words, but "Joe" ends in a vowel, which makes it better suited to being sung on an extended note, as in "Hey Joe." In "Farmer John," the name is sung in a short staccato syllable.
Connotations: "John" and "Joe" are both incredibly common names, with variants in use all across the world. "John Doe" and "Joe Bloggs" are both generic names used to mean an everyman. But John is an actual stand alone name, while "Joe" is typically a nickname. "Joe" is also often used in playful coinages like "Joe Cool." That makes "Joe" seem a bit more casual and intimate. "John" seems, to me, to be a bit more of a solid, stolid citizen --more of a farmer than a crime-of-passion murderer, if you will.
The connotations of the sounds: "John" is sometimes used to mean a toilet, and is also sometimes used to mean the customer of a prostitute. I don't think either of those connotations comes into play here. I'm not aware of any common alternate meanings for "Joe." So this seems like a bit of a dead end.
In summary, I'd say it's "Joe", not "John," partly because of the more apt sound of the name (for singing), and also because "Joe" is more casual and unpredictable than solid citizen "John."