This is the scene where John "Trapper" McIntyre is made Chief Surgeon.
I believe the other fellows were parading him around in celebration?
They sing a song about him being a king, and state he needs a queen.
To this end he jokes that he'll take Houlihan as his queen by pointing at her.
Hawkeye feigns misinterpretation and suggests he was pointing at Frank Burns.
This offends both Houlihan and Burns, who leave in a huff.
Duke calls out an insult as they leave that suggests Frank Burns has forgotten his shingle:
- Informal. A small signboard, as one indicating a professional office: After passing the bar exam, she hung out her shingle:
I believe this is part of the running joke where all the doctors question Frank Burns' skill.