The Australian and New-Zealand phrase to be a box of birds, and variants, mean to be in good spirits, ‘chirpy’. Gerald Alfred Wilkes (1927-2020) wrote the following in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990):

box of birds Equivalent to ‘in good spirits’: N.Z. rather than Australian.

The image is of a boxful of chirping birds—cf. in quotation 5 below, the extended form happy as a bird in a box of birdseed.

The earliest occurrences of the phrase to be a box of birds and variants that I have found are:

1-: From On the Mercy of the Court: A Short Story by “Nineteenth”, published in the Dalby Herald (Dalby, Queensland) of Friday 25^th August 1939:

“You know I was out at a party the other night,” she giggled, “and Bruce Pepper said to me—do you know Bruce? He’s such a box of birds—well, anyway, he said to me [&c.]”

2-: From The Sunday Mail (Brisbane, Queensland) of Sunday 12^th July 1942:

VIVID WAR SLANG

By Sidney J. Baker, who is compiling a dictionary of Australasian English, and who has written two books on slang.
Australian servicemen in this war are building up as rich and as varied a slang vocabulary as their predecessors of the last war.
[…]
Naval Terms
[…]
To feel like a box of birds: To feel happy, chirpy, full of beans.

3-: From That Summer, a short story by the New-Zealand short-story writer and novelist Frank Sargeson (Norris Frank Davey – 1903-1982)—as published in The Penguin New Writing (London: Penguin Books, 1943):

‘Hello Terry,’ he said, ‘how’s things?’
‘A box of birds,’ Terry said.
[…]
Well, you mightn’t believe it, but I woke up early feeling just like a box of birds.

4-: From The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945), by Sidney John Baker (1912-1976):

to feel like a box of birds, to feel happy or “chirpy”.

5-: From the caption to the following photograph, published in Truth (Sydney, New South Wales) of Sunday 12^th January 1947:

Happiest man at Randwick yesterday was owner J. Cloros. He’s owned only one gee-gee, and that’s Prince Mohican. And with three wins in a row he’s happy as a bird in a box of bird seed.

6-: From an interview of Nancy Wake, by Ruth Park, published in The Herald (Melbourne, Victoria) of Friday 18^th November 1949—born in Wellington, New Zealand, Nancy Wake (1912-2011) was a prominent figure in the French Resistance during the Second World War:

Nancy Wake has, as a political candidate, come in for a lot of publicity as the woman who broke German sentries’ necks in the late war, but actually she doesn’t look anything like that.
She doesn’t speak, either, like a man with a beard and a heavy cold. Despite her harsh experiences she is as bright as a box of birds, laughs easily, and is lots of fun.

7-: From Jimmy Brockett: Portrait of a Notable Australian (London: Britannicus Liber, 1951), by the Australian short-story writer and novelist Dal Stivens (1911-1997):

When we set out this Sunday morning in the Russell for Sadie’s people’s place at Mosman, I was feeling as happy as a box of birds.

A New-Zealand variant of the phrase is to be a box of fluffy ducks, also to be a box of fluffies. The earliest occurrence that I have found is from New Zealand on foot, by Cecil Kuhne, published in The Tampa Tribune (Tampa, Florida, USA) of Sunday 9^th September 1990:

New Zealanders have made slang a national art.
An “alpine stick” is a hot dog; “big bickies” is lots of money; “bubble and squeak” is a vegetable hash; a “goggle box” is a television; and “berk” is a jerk. An eccentric is sometimes said to be “as silly as a two-bob watch,” and a person doing well might be heard to remark, “I’m a box of fluffy ducks today.”
If someone says he’s going to “tingle you up,” don’t be concerned; it means he’s going to call you on the telephone. One who is angry is “cheesed off” or “got his knickers in a twist.”

Max Cryer recorded box of fluffies—and box of budgies *—in The Godzone Dictionary of Favourite New Zealand Words and Phrases (Auckland: Exisle Publishing Limited, 2006).

*The noun budgie is a colloquial abbreviation of budgerigar, denoting a small Australian parrot—cf. ‘budgie smugglers’: meaning and origin.

The variant box of fluffies occurs in Hunter’s Mark, independently published in 2018, by the New-Zealand author Wendy Smith:

“Hey, you old bitch. How are you?”
Despite her obvious tiredness, she smiles. Mum and I have always had an open and honest relationship. Or at least, I thought so until I discovered the line of crap she’d fed Adam for years. It’s the one thing I know of that she kept from me.
“A box of fluffies. Isn’t that what the Kiwis say?”
I sit on the side of the bed and lean over to kiss her temple. “You’re as much of a Kiwi as I am, Mum.”