The script of the 1972 "Free to be you and me" sketch is below, as the puppet clip missed out some of the lines. They really did have a better clue about sex equality and stupid stereotypes nearly half a century ago.
Mel Brooks (male, deep voice): Hi!
Marlo Thomas (female, higher voice): Hi!
Male: I’m a baby!
Female: Well what do you think I am, a loaf of bread?
Male: You could be, what do I know, I’m just born, I’m a baby, I don’t even know if I’m under a tree or in a hospital or what, I’m just so glad to be here.
Female: Well, I’m a baby too.
Male: Have it your own way, I don’t want to fight about it.
Female: What, are you scared?
Male: Yes, I am, I’m a little scared. I’ll tell you why. You see, I don’t know if I’m a boy or a girl yet.
Female: What’s that got to do with it?
Male: Well, if you’re a boy and I’m a girl you can beat me up! You think I want to lose a tooth on my first day alive?
Female: What’s a tooth?
Male: Search me, I’m just born, I’m a baby, I don’t know nothing yet!
Female: You think you’re a girl?
Male: I don’t know, I might be. I think I am. I ‘ve never been anything before. Let me see, let me take a little look around. Hmm… cute feet, small, dainty, yup, yup, I’m a girl, that’s it, girl time.
Female: Well, what do you think I am?
Male: You, that’s easy, you’re a boy!
Female: You sure?
Male: Of course I’m sure. I’m alive already four, five minutes, right? I haven’t been wrong yet.
Female: Gee, I don’t feel like a boy.
Male: That’s because you can’t see yourself.
Female: Why, what do I look like?
Male: Bald. You’re bald, fellah. Bald, bald, bald, you’re bald as a ping-pong ball, are you bald.
Female: So?
Male: So, boys are bald and girls have hair.
Female: Are you sure?
Male: Of course I’m sure. Who’s bald, your mother or your father?
Female: My father.
Male: I rest my case.
Female: Hmm. You’re bald too.
Male: You’re kidding!
Female: No, I’m not.
Male: Oh, don’t look!!
Female: Why?
Male: Ugghhh. A bald girl. Yuck. Disgusting.
Female: Maybe you’re a boy and I’m a girl.
Male: There you go again. I told you, I’m a girl. I know it, I know it, I’m a girl, and you’re a boy.
Female: I think you’re wrong.
Male: I am never wrong! What about shaving?
Female: What about it?
Male: You just shaved, right?
Female: Wrong.
Male: Exactly! And you know why? Because everyone’s born with a clean shave. It’s just that girls keep theirs, and boys don’t.
Female: So what does that prove?
Male: Tomorrow morning, the one that needs a shave, he’s a boy.
Female: Well I can’t wait until tomorrow morning!
Male: See, that proves it! Girls are patient, boys are impatient.
Female: Yeah, what else?
Male: Can you keep a secret?
Female: Absolutely.
Male: There you go! Boys keep secrets, girls don’t.
Female: Hmm.. Go on.
Male: Are you afraid of mice?
Female: No.
Male: I am, I’m terrified of them! I hate them! Squeak, squeak, squeak! What do you want to be when you grow up?
Female: A fireman.
Male: What did I tell ya?
Female: How ’bout you?
Male: A cocktail waitress! Does that prove anything to you?
Female: Hmm… You must be right.
Male: I told you, I’m always right. You’re a boy, and I’m the girl.
Female: I guess so. Ooh! Wait, here comes the nurse to change our diapers.
Male: About time too, I’ve never been so uncomfortable in my life.
Female: Hey, look at that!
Male: What?
Female: You see that? I’m a girl, and you’re a boy!
Male: Hey, it sure looks like it.
Female: What do you think of that?
Male: I can’t understand it.
Female: Well, it sure goes to show ya.
Male: What?
Female: You can’t judge a book by its cover.
Male: Ha, ha, ha. What does that mean?
Female: How should I know? I’m only a baby.
Male: So am I. Goo.
Female: Goo.