Nice vintagey eBay snag. These are oversized (3 x 5.5) and very thin. From what I can gather, National Chicle Gum Company was selling football cards in retail stores in 1935, baseball cards in 1936, and bankrupt by 1937. So many of these were chucked into the dumpster with empty boxes and old mop heads, I'm sure. This is my second from the set.
As for the two pictured, Chuck Dressen played for the Reds (1925-1931) and Giants (1933) and managed the Reds (1934-1937), Dodgers (1951-1953), Senators (1955-1957), Braves (1960-1961), and Tigers (1963-1966), amassing 1008 wins and two NL pennants. Bill Myers played 6 seasons for the Reds (1935-1940) and 1 season for the Cubs (1941) and was the starting shortstop for the Reds' World Series teams in 1939 and 1940 and was inducted into the Reds Hall of Fame in 1966.
Considering the signature misspells Myers' name, I'm going to go out on a limb that these signatures do not represent what their real ones look like.