Green’s Dictionary of Slang

booai n.

also booay, boohai
[? Maori puhoi, dull, slow or Puhoi, a failed mid-19C utopian settlement]

(N.Z.) the backwoods, remote rural areas.

[NZ]H.P. Simpson If You’d Care to Know 101: Another New Zealand expression is ‘Up the Bouai’ (pronounced boo-eye). There was a place near Auckland city which many years ago was somewhat isolated, and it was called the Bouai. [...] To go ‘up the Bouai’ has become fairly general in use. Today, if a person comes to town from any lonely place in the forests or mountains, and a friend asks him where he has been, most likely the answer will be, ‘Oh, I’ve been up the Bouai.’ It is a general word for an isolated region [DNZE].
[NZ] (ref. to c.1922) R. Mason postcard in DNZE (1998) 77/3: boohai heard and used in Pukekohe, Auckland, c 1922 .
K. Sinclair Hist. of N.Z. 97: The Bohemians who founded Puhoi (the name of which, corrupted, is apparently the origin of the slang term ‘the Boo-ay’, a synonym for ‘the out-backs’).
[NZ]R.L. Bacon In the Sticks 184: sticks — ‘out in the sticks’, ‘out in the backblocks’, ‘out in the bush’, ‘out in the booay’, ‘out in the cactus’, all have the same basic meaning of ‘fifty miles from nowhere’ [DNZE].
[NZ]U.T.C. Song Book (3rd edn) Song No. 36: It’s a long drive up from the buwai By Woodcocks and Kairpara [sic] flats [DNZE].
[UK]Listener (NZ) 19 Apr. 56: Well, out of the boo-ai comes three trampers. One was a nuggety bloke in a sou’-wester [DNZE].
[NZ]N.Z. Herald (Auckland)1 Mar. 1: The oldest son, John, left Puhoi [c 1900] when he was 16 to settle in Auckland [...] ‘The expression “up the Buhoi” was said to have originated with him because when he was asked where he came from, that was what he replied in his strong accent – the same pronunciation of the name of our village can still be heard in Puhoi today,’ said Mrs Williams [DNZE].
[NZ]G. Lay Fools on Hill 220: Living in a shack in the middle of the boo-ai with a load of strangers, doing the washing by hand under a tilly lamp? [...] it wouldn't be what I'd call freedom.

In phrases

up the boohai (adv.)

(N.Z.) totally confused, absolutely wrong.

[NZ]W.J. Morrell (Wanganui) quoted in letter 30 Apr. 1984 to Heinemann Dict. of N.Z. Quotations n.p.: ‘Up the booeye for the rhubarb season’. Saying of the 1930s. Source unknown [DNZE].
[NZ]B.J. Cameron Collection (TS July) n.p.: bouai up the bouai (adv.) Up the spout [DNZE].
[NZ]G. Slatter Gun in My Hand 91: Got the pricker with me. Slingin off at me he was. You’re up the boo-ay he told me.
[NZ]R.M. Muir Word for Word 171: ‘She’ll be right up the boo-ay if we’re not lucky,’ Arthur said.
[Aus]G.W. Turner Eng. Lang. in Aus. and N.Z. 108: New Zealand has some slang of its own [...] Up the boo-ay means either ‘altogether wrong’ or ‘out in the backblocks’.
[NZ]McGill Dict. of Kiwi Sl. 18/1: boohai/boo-ay/boo-eye remote area; probably corruption of Maori word Puhoi, the North Auckland Bohemian settlement so isolated early on its pioneers almost perished from hunger; phr. up the boohai to be lost or to have got something quite wrong, a nod perhaps at ‘puhoi’, Maori for dull, slow, phlegmatic; up the boohai – shooting pukakas (pukekos) an amusing extension of notion of being lost, possibly in the head, Sue Budd recalls. Grant Tilly recalls its completion: ‘up the boohai shooting pukakas with a popgun.’.
[NZ]McGill Reed Dict. of N.Z. Sl. [as cit. 1988].
way to boohai

(N.Z.) intensifying remoteness or distance; ‘way to blazes’.

[NZ]C. Hunt Speaking a Silence 88: When we got there, way to boo-eye up Lake Rotoiti way, we had to shovel the snow away before we put the tent up [DNZE].