The look and style of the neckerchief is an individual unit decision.
According to the BSA's Guide to Awards and Insignia (see page 12), the option to wear the neckerchief over or under the collar (or even to wear a neckerchief at all) is up to individual units to vote on. Boy Scout neckerchiefs are optional and troops choose their own official neckerchief colors and style. Additionally, according to the guide: "official neckerchiefs are triangular in shape" and "the neckerchief is worn only with the official uniform and never with T-shirts or civilian clothing."
A good neckerchief is a great way for building unit identity, unity and pride. As Baden-Powell said, "every Troop has its own scarf color, and since the honor of your Troop is bound up in the scarf, you must be very careful to keep it clean and tidy." A neat and sharp-looking neckerchief is a great way to look sharp and show Scout spirit as a unit.