What does the phrase "shake a chicken (in the middle of the room)" mean?

'Shake a chicken' is probably a reference to the dance style, the Chicken

The Chicken was a style (or styles*) of dance:

The Chicken is a popular rhythm and blues dance started in America in the 1950s, in which the dancers flapped their arms and kicked back their feet in an imitation of a chicken. The dance featured lateral body movements. It was used primarily as a change of pace step while doing the Twist.

Confusingly, it is different from the similarly-named Chicken Dance (also known as Bird Song or Chicken Song), which features accordion music and is likely the better known. A recording of this song, The Birdie Song charted in the UK in 1981.

Given the content of the song involves preparations for, and various actions at, a party it is reasonable to assume that the lyrics refers to a dance which was contemporaneously gaining popularity in the United States.

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^{*: Another source claims that "The Chicken is actually the name of several completely different dances. According to Manny Interiano of the “KPIX Dance Party”, it was basically a swing dance/cha cha variation."}