Private contractors in Kuwait are being blamed for the mistake, and Australian Prime Minister John Howard said he feels "desperately sorry" for Mr Kovco's family.
The body of Private Jake Kovco, 25, was due to arrive at Melbourne airport early today ahead of a funeral with full military honours, but the wrong casket was loaded onto the plane.
"It's an absolute disgrace them losing the coffin. They promised that he would have someone with him all the time," Pte Kovco's cousin Adam Backman told Southern Cross Radio.
"It's shocking. It's a disgrace. The government and the Australian army should be ashamed of themselves.
"John Howard, you're nothing but a shocker."
Initial reports said that Private Jake Kovco died when he accidentally shot himself while cleaning his service pistol in Baghdad on Friday.
But Defence Minister Brendan Nelson today said the reports were incorrect.
"The family's actually quite distressed because the reports are that he was cleaning his weapon," the minister told Macquarie radio.
"He wasn't in fact cleaning his weapon. It was near him in his vicinity and he made some kind of movement which suggests that it discharged.
"There was obviously a live round in it which there should not have been and that's as much as I should probably say at the moment."
Mr Backman said the family is angry over the misinformation.
"We need closure to this. I can see the way everybody is talking, the government and everything, we're never going to be told the truth about what happened to him," he said.
"The story changes and changes and changes."
Wrong casket
Private Kovco’s grieving family was at Melbourne airport on Thursday morning waiting for the return of his remains, but after a mistake apparently made during a stopover in Kuwait, the wrong casket was flown to Australia.
Dr Nelson has expressed his regret at the mix-up, but stopped short of apologising for the bungle.
He said a private mortuary in Kuwait appears to be responsible for the mistake.
"I am not in a position at the moment to express sorrow, if you like, on behalf of the Australian government, and let's just find out what the cause of it was first," Dr Nelson told reporters at Melbourne Airport.
"Let's get him back and then let's find out what happened before we start assigning blame."
Private Kovco's mother Judy also spoke of her heartbreak over the events at Melbourne Airport, when the bungle became apparent.
"I was waiting for my son to walk through the door," she told The Age newspaper.
"I wake up each morning thinking it was a nightmare. I was just devastated my boy wasn't walking through that door. I realised he wasn't going to come through there."
Pvt Kovco's body is now expected to return very early on Saturday morning.
"Double whammy"
Australia's defence force chief, Air Marshal Angus Houston, described the bungle as a double whammy for his family.
"I'm very upset about it ... what has happened is totally unacceptable," Air Marshal Houston told Southern Cross Broadcasting.
"What we are doing is essentially everything to ensure something like this never happens again."
The private company responsible, Kenyon International, told ABC radio it is still unsure how the mistake occurred.
The company has been involved in the recovery of bodies from more than 300 disasters, and a spokesman said they are investigating what happened.
Spokesman Mario Gomez said he does not know the identity of the body mistakenly sent to Australia, although the government said it is that of a soldier from an eastern European nation.
ALP ire
The federal opposition has lashed out at the government, saying it must ban private contractors from handling the bodies of Australian military personnel killed in the line of duty.
"I'm not going to seek to politicise this. All any Australian feels is just simply the utmost sympathy for Shelley Kovco and her family," said opposition leader Kim Beazley.
"Never again must any Australian casualty be handled by private contractors. Never again must circumstances occur where our honoured dead are brought home by anyone other than the services for whom they fought."
Money appeared to be the motivation for using private contractors, he said.
Funeral delayed
The 25-year-old private was a member of the 110-strong security detachment in Iraq's capital to protect Australian officials.
The federal government is now considering chartering a private plane to bring him home. It was informed of the mistake on Wednesday night.
Mrs Kovco asked to speak to Mr Howard who was woken about 11pm so that she could express her anger.
Dr Nelson said a senior defence force official and an independent pathologist would be despatched to the Middle East next week to investigate the mistake.
Army helping family
Chief of Army Lieutenant-General Peter Leahy said the army was helping Private Kovco's family, including his two children Tyrie, four, and Alana, 11 months.
"... a commanding officer from Jake's battalion is with the family now, along with his mates," Lt-Gen Laehy told reporters.
"Shelley asked last night to be with them and they're supporting her, her family and her children as much as they can and we'll make sure we provide them all the support we can over these next few difficult days."