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
You need not pay at once.
You do not need to pay at once.
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
You don't need to pay right away.
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.