Bouquet was born in Rolle, Switzerland, in 1719. He was likely the son of Isaac-Barthélemy Bouquet, a mercenary in the service of the Kingdom of Sardinia, and Madeleine Rolaz. Like many military officers of his day, Bouquet traveled between countries serving as a professional soldier. He began his military career in 1736 in the army of the Dutch Republic, and later in 1739 joined the army of the Kingdom of Sardinia. In 1748, he was again in Dutch service as lieutenant colonel of the Swiss guards.
