The Australian National Dictionary Centre (Australian National University, Canberra, Australian Capital Territory) defines and explains the Australian-English phrase may your chooks turn into emus and kick your dunny down as follows:

A jocular curse. This expression recalls an earlier time when many Australians kept chooks (domestic chickens) in the backyard and the dunny was a separate outhouse.

The earliest occurrences of the phrase may your chooks turn into emus and kick your dunny down and variants that I have found are as follows, in chronological order—the phrase was popularised, if not coined, by Barry Humphries [cf. note]:

1-: From Bazza arrives on the silver screen, by ‘Batman’, published in The Bulletin (Sydney, New South Wales) of Saturday 19^th August 1972:

At last we have a film […] that shows off the genuine Australian in a well-rounded, exhilarating light—“The Adventures of Barry McKenzie.” [cf. note]
As you know Bazza McKenzie is the creation of Barry Humphries […].
[…]
Barry Crocker as Bazza is beautiful. He shows off remarkable cultural achievements, like being able to open a tube of the chilled article with one hand and give himself a swift amber transfusion with the speed of light. He wears the big hat, the double-breasted R. G. Menzies ¹ suit and he is utterly unlike any Australian in Earls Court today. He is the mythical Australian seen through the eyes of The Sunday Times.
His language too has a curious Frank Hardy ² flavor, containing sufficient ingenuity to take over the world. Some examples: “She bangs like a shithouse door in a gale,” “I’m as dry as a nun’s nasty,” “I’m as dry as a dead dingo’s donger,” or “I hope your chooks turn to emus and kick down your dunny.” And when an arrogant Pom asks Bazza about his convict origins he disarmingly replies: “Go and dip your left eye in cocky shit.”

¹ This refers to the Australian politician Robert Gordon Menzies (1894-1978).
² This refers to the Australian author Frank Hardy (1917-1994).

Note: The Adventures of Barry McKenzie (1972) is an Australian comedy film directed by the Australian film director Bruce Beresford (born 1940), written by Bruce Beresford and the Australian comedian Barry Humphries (born 1934), and starring the Australian comedian Barry Crocker (born 1935). This film tells the story of an Australian yobbo on his travels to the United Kingdom. Barry ‘Bazza’ McKenzie was originally a character created by Barry Humphries for a cartoon strip in the British satirical magazine Private Eye. (Barry Humphries also created and interpreted Dame Edna Everage, a fictional character satirising the average Australian housewife.)

2-: From the review of The Adventures of Barry McKenzie, by Kelvin Rowley, published in the Tribune: Australia’s Communist Weekly (Sydney, New South Wales) of Tuesday 28^th November 1972:

This is for the most part a very funny film. I doubt if anyone but Australians would find it so, if only from the fact that some of the best lines would be incomprehensible. Why is it so funny?
The main element in its humor is the dialogue, the repartee (if that’s the right word) which carries the sardonic Australian slang through to its logical absurdity.
Consider: “As dry as a dead dingo’s donger” or “may all your chooks turn into emus and kick down your dunny”, or “go and stick your head up a dead bear’s bum”, etc., etc, Barry’s witticisms are like much Australian humor—either urinatory, anal or genital.

3-: From Brushing up ‘Bazza’ for UK audiences, a correspondence from London about the release in the United Kingdom of The Adventures of Barry McKenzie, published in The Sydney Morning Herald (Sydney, New South Wales) of Wednesday 21^st February 1973:

Humphries gave some examples of how sub-titles will be used in the “English” version of the film.
Where Barry McKenzie says to an upper-class Englishman in one scene: “I hope your chooks turn into emus and kick down your dunny,” a translation line will appear on the screen saying: “Destroy your outdoor toilet.”

4-: From a letter published in Tharunka (Kensington, New South Wales) of Wednesday 21^st May 1975—Tharunka is a student magazine published at the University of New South Wales:

Dear Editors,
I realise that this letter will probably not be read and will definitely not be printed in your FREE press, but I feel that it is my duty to get off my chest what has been giving me the grizzlies for weeks.
[…]
Right—go ahead and tear up this letter, at least you know I said it, and I mean it! To you goddamn editors: “May the Bird of Paradise fly up your nostrils”. “May your chickens grow up into emus and kick your dunny door down” and “May the dead horse you’ve been flogging all these years jump up and fuck hell out of you.”!
Yours maliciously
Melba (an art fart)

5-: From May we quote you?, published in The Age (Melbourne, Victoria) of Saturday 28^th June 1975:

I HOPE all Mr. Prowse’s chooks turn to emus and kick his dunny down.
—Barry Crocker on the banning of The Adventures of Barry McKenzie from Australian TV by the Chief Film Censor, Mr. R. J. Prowse.

6-: From ‘Every country has its Bazzas’, published in The Canberra Times (Canberra, Australian Capital Territory) of Monday 8^th September 1975:

SYDNEY, Sunday.—Barry Crocker, the Australian singer who played the part of Bazza McKenzie in the films, said today that every country had its “Bazzas”.
[…]
He was being interviewed at Sydney Airport on his return from Las Vegas where he appeared at nightcubs [sic] for a month.
Mr Crocker said Bazza “reared his ugly head” only a few times in the US, because whenever he did the Americans did not seem to understand what the character was all about.
“Once I was asked to rattle off a typical Bazza catch-phrase, so I said ‘I hope all your chooks turn into emus and kick your dunny down’, which was greeted with a blank expression”, he said.
“If that was translated into an Americanism, it would probably sound something like, ‘I hope all your chickens turn into very large birds and kick your bathroom over’, which sort of loses all the flavour”.

The phrase may your chooks turn into emus and kick your dunny down was also “translated” in the following two texts:

1-: In Watch your language, by Roger Prior, “Resident Aussie of the Times”, published in The Times (Nanaimo, British Columbia, Canada) of Thursday 10^th July 1980—here, trans is short for trans-Pacific:

AUSSIE: If you’re not dinkum your chooks will turn into emus and kick your dunny down.
TRANS: If you don’t tell the truth your domestic hens will turn into emus (giant Australian birds) which will demolish your outside privy.

2-: In The gurus of gobbledegook, by Phillip Adams (born 1939), the producer of The Adventures of Barry McKenzie, published in The Bulletin (Sydney, New South Wales) of Tuesday 12^th August 1980:

When we released The Adventures of Barry McKenzie in London, the ocker argot fell on deaf ears. So Barry Humphries decided to aid the Poms’ perception with sub-titles. These reached their apotheosis at the Young Conservatives’ dance at Rickmansworth where a toffy-nosed Pommy Bastard kept picking on our Bazza. And every time he’d say something like, “Hello Cobber Digger, what’s it like walking around on your head?” the antipodean Candide would lash back with, “I hope your chooks turn to emus and kick your dunny down.” This, of course, could be sub-titled as “May your buff-orpingtons be transformed into large, flightless birds which proceed to vandalise your WC.”
British audiences found it even harder to come to terms with the next verbal exchange: “Go and stick you head up a dead bear’s bum.” But once rendered into “Go and thrust your cranium up the sphincter of a deceased bruin,” Bazza’s meaning was abundantly clear.