Every so often on 'Burke's Law', Captain Amos Burke threatened to fire his Asian man-servant, but Henry always knew that it was just a bluff.
In "Who Killed Carrie Cornell?", Captain Burke snapped at his chauffeur: "I'll give you back to Charlie Chan!"
He could have done it, too. The legendary Chinese-American detective was played by J. Carroll Naish in 'The New Adventures Of Charlie Chan' in 1957.
So if we take that threat of Burke's at face value, then it's possible that he hired Henry away from Charlie Chan!