Are these two sentences equivalent?
You needn't pay at once.
You don't need to pay at once.
If yes, which one would you recommend? Is it an US/GB thing?
They are equivalent in meaning; however, the non contracted forms would be
I think the first is more common in BrE (though I would request confirmation). The second formation is more common is AmE; however, we would more likely say
I initially just thought needn't is probably more British usage, but that it's becoming increasingly archaic / affected.
So I produced this NGram to support my thinking. Restricting to just American or British doesn't suggest it's much more common in either.
Frankly, I just don't know what to make of this one showing the latest trend.
Nevertheless, I'd still advise OP to use don't need to. I doubt anyone would think that meant he wasn't keeping up with the times.
We can both use need not and don't need to. However, if needn't is followed by an object, we must use don't need.
"Coat" is an object, so it is wrong to say ,"You needn't your coat".