John Barlow, press agent for the play, to be directed by Alan Bailey, said Tuesday it will open at the Acorn Theater on West 42nd Street July 29 for a 13-week engagement.
He said the play, in which Randolph performs, deals with Randolph's experiences while working in a woman's homeless shelter, her love affair with an accordionist, her discovery of dignity and grace in unusual people, and finally self-acceptance.
Bancroft saw "Squeeze" in Los Angeles more than two years ago and said, "It was absolutely love at first sight -- like when I first met Mel Brooks," according to Barlow.
Brooks is Bancroft's husband and the author of the long-playing Broadway comedy, "The Producers." Bancroft was quoted as saying she had learned a lot from her husband's show, notably "What not to produce."