New American Puzzle

Good quality plywood jigsaw puzzles which are thought to be later than the Modern branded puzzles.
Boxes also feature a Manufactured by entry on the label. The first three examples I have been shown all had J.A.B. as manufacturer, but I have since been made aware of other examples with B.B.Co., recorded as being the maker.
Any help with identities for these sets of initials gratefully received. Many thanks.