American show hunters are a whole different thing than the UK working hunter. The UK working hunter has stayed a lot closer to its roots as a field hunter, the American show hunters has evolved into a discipline all its own with its own conventions and criteria. Most American show hunters would have a case of the vapors and collapse if you attempted to fox hunt them, they are "hot house flowers." Also, I've seen video of the winning horses at HOYS; they are lovely horses that are forward, brave and clever jumpers, but they would not pin in an American show ring because they lack the style and way of going expected.
Google "John French and Rumba" or "Destry Spielberg and Rumba" (can't post links from work, sorry) or look at the link posted earlier in this thread and you'll get an idea of what's pinning in the American hunter ring. Oh, and those horses? They are considered to show a lot of pace and expression as opposed to a normal Children's or Adult Amateur class.
To the OP, the flat jumping style of an equitation horse is not something that's really trained for initially. If a horse is builit to go that way, or learns to go that way defensively with a restrictive rider, then they are developed as eq horses, where as round, scopey jumpers as slotted towards hunters. Just train your horse to be a tidy, careful jumper.
And I'm still not sure I know what you mean by "hunter/jumper"