'Leg-orn' really is the proper pronunciation. 'Leggern' is closer to proper than 'leg-horn,' though.
The breed is named after the Italian port city Livorno. Way back when, Livorno was called 'Legorno' in English, from which we dropped the -o ending and ended up with Legorn. Somewhere along the way, a silent h was added in the middle. From that traditional English name of the city, we got the name for the chicken breed as well, Leghorn. The h should be silent in the
proper pronunciation of Leghorn, much in the same way that we don't pronounce the h in spaghetti. And no red-necking was necessary to come up with that explanation.
I believe the pronunciation difference is less geographical and more a matter of how you've heard it said. A whole slew of people, Americans especially, have grown up with an awareness of the character Foghorn Leghorn, whose name was mispronounced intentionally for the reference to Senator Claghorn (the character that Foghorn parodied), and somewhat as well for the implication that he isn't exactly the sharpest tack in the box. Unfortunately, more people know of Foghorn than they know of the chicken breed, or have heard the name Foghorn Leghorn before they learn of the chicken breed, so the breed largely gets mispronounced as a result. At least, that's the conclusion I've drawn from the varied responses that come from this question being asked over the years.