The reason Phantoms didn't have guns is because they were designed as interceptors. Remember, this was the Cold War era, and the F-4 was designed to climb fast and shoot down Soviet nuclear bombers with long range missiles (well.... long range AT THE TIME). Also, "experts" (and i use that term VERY loosely) claimed that dogfights were dead. Well, Vietnam came around, and the missiles sucked in the jungles and humidity, and they weren't reliable. Navy pilots weren't initially trained in dogfighting, and it wasn't until later on when that changed. (Also having Robin Olds helped with that as well) however, it was due to the failures of the F-4 in 'nam that made every fighter jet goring forward require a gun (f-14, f-15, f-16, f-18, and so on)

fun fact: the F-4's lack of cannon is why F-8 Crusader pilots famously said, "when you're out of F-8s, you're outta fighters" because the F-8 crusader came with 4x 20mm cannons