"Esquire" entered English because of the Norman French invasion of England in 1066. It means "bearer of a shield" in Old French, and was a necessary title for those Norman French in England who were not direct nobility, but needed to distinguish themselves from the common English peasantry. An "Esquire" was a servant to a knight, akin to a golf caddy, who carried the knight's shield.
As the French influence faded in medieval England, mere English landowners adopted the term (modified to "Squire"), and it became a term used for minor "Lords of the Manor" who owned the majority of the land in a village, in order to indicate their superior social status.
The codification of titles in England led to this 'title' becoming official for someone of status who was not a "Knight of the Realm", and it was abbreviated as "Esq." in publications such as Burke's Peerage, eventually becoming associated by the 18th Century with the professional classes, such as lawyers. Someone designated "Esq." was above the wastrel descendants of the upper classes, who were merely "gentlemen", but not true nobility. It was at his point in English history that the term was exported to the USA, where it remains to this day associated mainly with lawyers and attorneys.
I know of no reason why there should be any distinction between full names and initials when using the honorific "Esq."