Dr. Chris Chun's name is a play on that of Dr. Christian, a character played by Jean Hersholt in a series of six moralistic but popular features released 1939-1941. The cartoon doctor is a caricature of Hersholt.
The switchboard operator pages Dr. Ehrlich and Dr. Cyclops. Films about the real Dr. Paul Ehrlich and the fictional Dr. Cyclops were released earlier in 1940 (Dr. Ehrlich's Magic Bullet (1940) and Dr. Cyclops (1940)).
"The Pains Came" is a medical pun on "The Rains Came", a 1937 novel and a 1939 disaster movie based on it.
Young Dr. Chill-daire is a pun on not only Dr. Kildare, but of Frigidaire appliances.
The nurse pages "Sir Gerry," a pun on Surgery.