Short answers, pulled from the story.
In British military history, a billet was originally a written note ordering a soldier to be housed with a specific civilian. The practice of quartering soldiers in private homes was outlawed in 1689 as an extension of a section of the Petition of Right 1628, following repeated petitions against it dating back to the 16th century.
The Third Amendment was a direct response to the British practice of quartering soldiers in colonial homes, which was one of the major grievances of American colonists leading to the Revolutionary War. It places restrictions on how the federal government may require civilians to provide housing for soldiers.
In the US Navy, Marine Corps, and Coast Guard, a billet is a specific personnel position or duty assignment that can be filled by one person. Every sailor or marine reporting to a ship or shore installation is assigned a billet according to the unit's watch, quarter and station bill.
In French-influenced parts of the world, including eastern Europe, billet became the word for a train ticket because military billets issued to soldiers specified their sleeping quarters on overnight trains. When trains were overloaded, only soldiers holding a billet were guaranteed a berth, making the document synonymous with a ticket.
A billet family in junior ice hockey is a family that provides room and board to young players who have left home to join elite teams in other towns. Coaches are typically involved in matching players to billet families, and the arrangement aims to give players a home away from home during the season.
Fears over child safety in Canadian amateur sports drastically curtailed the billeting practice. Many clubs stopped billeting entirely, while others introduced mandatory criminal record checks for coaches, volunteers, and anyone over eighteen from a host family, working through their provincial sports bodies.