He has the hiccups. This means he's suffering from a case of the hiccups.He has a hiccup.
He's got the hiccups. He got the hiccups.He get a hiccup.
This, while unusual, could work if you mean his hiccups are generally loud, like "He has a loud voice". But if you mean his hiccups are loud on a certain occasion, as I think you do, I suggest His hiccups are loud or He's hiccupping loudly.He has a loud hiccup.
But JulianStewart's (#7) link to the Ngram Viewer shows that it's not a BE/AE thing—it's an old/new thing.I wish I thought most AE speakers would pronounce it correctly (or even understand it).