His comments came after a United Nations report - backed by the UN Secretary General Kofi Annan - called for the immediate closure of the controversial site in Cuba.
Mr Hain was believed to be the first Government minister explicitly to demand the closure.
He also said he believed that Prime Minister Tony Blair shared his view.
Mr Hain said on BBC1's Question Time: "I would prefer that it wasn't there. I would prefer it was closed, yes."
Asked if it was Government policy that Guantanamo should be shut down, he replied: "That's what I think."
And challenged on whether Mr Blair agreed with him, he said: "I think so, yes."
The camp was opened in 2002 to hold terror suspects seized during the Afghanistan war and is currently believed to contain around 500 inmates.
Mr Blair told MPs last November that Guantanamo Bay was "an anomaly that sooner or later has to be dealt with".
Mr Hain said that the British Government accepted that useful information had been obtained from detainees at Guantanamo, but had always been uncomfortable with its existence.
"What we've said all along is, we don't agree with that," he said last night. "(The Prime Minister) has said that, as a matter of fact, some of the information that came from there was of importance, but that doesn't mean to say that he thinks the place should have been set up in the first place. There's a distinction there.
"We've made clear our position. We've made it absolutely clear. (Foreign Secretary) Jack Straw's made it clear, so has the Prime Minister."
He added: "We've always said that Guantanamo Bay was something that shouldn't have happened."
Yesterday's UN report, ordered by the body's Commission on Human Rights, called on the US government to refrain from any practice "amounting to torture or cruel, inhuman or degrading treatment" at Guantanamo.
It said all detainees should be brought to trial or released "without further delay" and the facility closed.
Some aspects of prisoners' treatment, including force-feeding hunger strikers, amounted to torture, it said.
Mr Annan said: "I think sooner or later, there will be a need to close Guantanamo, and I think it will be up to the (US) Government to decide, hopefully, to do it as soon as is possible."
Following the report's publication, the US administration dismissed its findings as "largely without merit".
The five UN investigators who compiled it had refused an invitation to visit Guantanamo Bay, said a spokesman, because the US would not allow them interview detainees.
They relied instead on interviews with former detainees, public documents, media reports, lawyers and a questionnaire filled out by the US Government.
Nine British nationals who had been detained there have now been flown back to the UK and released without charge.
None of the current inmates at Guantanamo are British, but Amnesty International believes that eight have previously been resident in the UK and that some have relatives here.
Yesterday, three long-term UK residents - although not British citizens - were given the go-ahead to seek a High Court order requiring Foreign Secretary Jack Straw to petition for their release from Guantanamo.
A judge in London said allegations of torture being practised at the facility meant the detainees, and their families living in the UK, had an arguable case that the British Government was under an obligation to act on their behalf.
Lawyers for the three men, Bisher al-Rawi, Jamil el-Banna and Omar Deghayes, and their families, were told there was "no guarantee" they would win the case, expected to be heard in full in mid-March.
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