The article indicates that this goes way beyond quartering. The police wanted to use the Mitchell home as an observation post to watch his neighbor (apparently, there was a domestic violence issue). They then decided to break into his home, assault him, arrest him and then take over the home. From the article:
The Mitchell family says that’s essentially what happened when Henderson police allegedly arrested them for refusing to let officers use their homes for a “tactical advantage” in a domestic violence investigation into a neighbor, according to an official complaint.
Police officers contacted Anthony Mitchell on July 10, 2011, with a request to use his house as a lookout while investigating his neighbor. When Mitchell told police that he did not wish to be involved, the complaint alleges, police decided they would use the residence anyway.
According to Courthouse News Service, the police department decided that if Mitchell refused to leave or open the door, officers would force their way in and arrest him.
Mitchell claims this is exactly what happened. First officers “smashed open” Mitchell’s door with a “metal ram” after he did not immediately open it himself. He then “curled on the floor of his living room, with his hands over his face,” as the police shot Mitchell and his dog — which the family claims did not attack the officers — several times with “pepperball” rounds.
Pepperball is a projectile containing chemical irritant pepper spray, which is released upon impact.
Afterward, Mitchell was arrested for “obstructing a police officer.”
The ordeal didn’t end there. Mitchell’s parents, Michael and Linda, were also neighbors to the home where police officers suspected domestic violence, so the police wanted to use their home as well. Michael Mitchell was invited to a local police command center to assist “in negotiating the surrender of the neighboring suspect.”
But upon arriving at the commander center, the elder Mitchell was informed the negotiations wouldn’t be taking place, the complaint says. When he decided to leave, he was also arrested.
The elder Mitchell’s wife was not arrested, but she was roughly escorted from her home while other officers entered the house without permission, the complaint alleges. The family claims that when she was allowed to return, “the cabinets and closet doors throughout the house had been left open and their contents moved about… Even the refrigerator door had been left ajar, and mustard and mayonnaise had been left on their kitchen floor.”
There's a whole slew of Constitutional rights violated here. The taking of the home violates the 5th and 14th, as cited elsewhere, but there is also a 4th Amendment violation. The text states that "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized." The forced entry, seizure of the home and arrest constitutes a violation of the rights of Mitchell. He was charged with "obstruction of a police officer", which is usually applied to persons who interfere with an arrest by attempting to physically hamstring the police. The arrest was completely trumped up. The detention of the father was also illegal, as he was not a suspect and had no obligation to be present for a false negotiation attempt (in fact, he had no obligation to be present for a real one, but chose to assist the police). The Henderson PD has a huge lawsuit coming, and the Third Amendment piece is a stretch, because while the definition of troops at the founding would certainly have included a police force as we now understand them, the question of whether they were "quartered " in the house may be an issue. I hope that the attorney is filing under multiple violations.