Were there any examples of Mercenaries or Private Military companies during WWI or WWII?

You might consider the American Volunteers Group, better known as the Flying Tigers.

While given clandestine authorization from FDR, the members were recruited and paid, along with supplied by aircraft from the Central Aircraft Manufacturing Company (CAMCO), which was organized by head sales rep of Curtiss(maker of the P-40 used by the men) in China, with funds funneled through a shell company by FDR's China guys.

The men were recruited and either resigned or took leaves of absence to be PMC's and were paid in general better than they had been in the service.