John Sheridan, 34, estranged wife Barbara and Mary Flynn, aged in her 70s, were offered council accommodation along with scores of other families from the site, before the massive operation.
They appealed at Southend County Court, which threw out the case, but appealed in the Court of Appeal at London's High Court.
The court judged the council had not acted unreasonably or unlawfully in offering the flats or interferred wit htheir human rights.
Appeal judges Sir Andrew Morritt, the Chancellor of the High Court, Lord Justice Patten and Lord Justice Pitchford.
They said there was "really no evidence" to support an argument that "possible sites" existed and a local authority could not be required to acquire land as part of a duty to provide accommodation for homeless people.
More in Thursday's Echo.