They argued in the High Court the house was part of land which Mr Murray farmed though the High Court judge said this was "preposterous" and they had "brought this on themselves".
They appealed and on Friday the Supreme Court upheld the High Court decision.
In their appeal, they argued, among other things, the demolition order was incompatible with their property, family, parental and private ownership rights under the Constitution (Articles 40.3, 40.5, 41, 42, and 43) and with similar rights under the European Convention on Human Rights.
Giving the Supreme's decision, Mr Justice William McKechnie said the court was mindful of the hardship it would cause to the Murrays, but it could not lose sight of the fact they had been living in an unauthorised development which was in flagrant breach of the planning laws for over a decade.
One of the principal matters which the court focused on was the issue of the inviolability of the family home.
Mr Justice McKechnie referred to a decision of then-High Court, now Court of Appeal judge Gerard Hogan in relation to a chalet built in a scenic part of Wicklow without planning permission.
That case related to Katie Fortune who lived in the chalet at Lough Dan, Roundwood, for 14 years with her two children.
In his 2013 judgment in that case, Mr Justice Hogan refused to grant a demolition order based on the home inviolability provision of the Constitution.
The Supreme Court said Mr Justice Hogan also drew on what was then another recent Supreme Court decision in relation to the requirement that search warrants must be sworn by an independent person rather than a garda officer close to an investigation.
That case related to Ali Charaf Damache (50), an Algerian-born Irish citizen living in Waterford, who was wanted in the US for an alleged terrorist conspiracy by an American Muslim convert, Colleen LaRose, known by her online name as "Jihad Jane".
She was in touch with Damache about her plan to travel to Sweden to kill artist, Lars Vilks, over a series of drawings depicting the prophet Mohammed as a dog. The plot was never carried out and she was later jailed.
Mr Justice Hogan found the findings in the Damache case meant the constitutional inviolability of a dwelling was "a free standing, self executing guarantee which applies to both civil and criminal proceedings and to both State and non-State actors alike".
Mr Justice McKechnie said he did not believe the Damache decision should be read in this way.
He was reluctant to endorse Judge Hogan's "free-standing self-executing guarantee" statement because it was in such non-qualified and unconditional language of "immense reach with a potential capacity to travel to destinations quite unknown".
Judge Hogan elevated the constitutional protections afforded by the inviolability provision (Article 40.5) to "too high a position of prominence" in the context of applications to demolish unauthorised buildings.
In this (Murray) case, Mr Justice McKechnie did not accept their submission that the High Court failed to give proper weight to the personal hardship a demolition order meant.
The Murrays were fully aware of the necessity to obtain permission and the Supreme agreed with the High Court's view what they did was "particularly flagrant and completely unjustified on any basis".
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