How did some americans justify their taking over spanish and seminole land in florida?

After the French and Indian war, the Treaty of Paris was signed, saying the British colonists won ownership to Canada, the Great Lakes, Ohio River valley, and Florida 

Spain was fighting a losing battle against revolutions in South America. Florida was mostly a vast swampland, and, being separated from the rest of the Spanish territory, it just caused a dispersal of military manpower. Added to the U.S. however, it would make borders tidier and more defensible, largely preventing, for example, the sort of north-south pincer movement the British tried in 1814.