I'm so happy to hear that the old-fashioned idea of a book-club or literature club is still alive! And that you are reading Shakespeare is even happier news! You GO, girls!!
In Shakespeare's day, the word "wanton" was used variously. It generally meant a female who doesn't follow the rules, so it tended most often to be used to denote women of ill-repute, prostitutes, or women who were said to be engaging in behavior bad enough to make them prostitutes.
Strangely enough, it was also used to denote little girls and teens who were merely spoiled and wanted their way. (This probably tells us a LOT about Shakespeare's culture's attitudes toward women's behavior, eh? Any kind of misbehavior could be assumed to lead to sexual misbehavior, it seems.)
So, this phrase, "a wonton's bird" literally means a little girl who is spoiled enough to have the luxury of a pet bird. The usual practice was to take a piece of string and tie the bird's feet to her finger or wrist, so that it wouldn't fly away. As you can imagine, this usually didn't result in very healthy conditions for the poor bird.
The image here is that Juliet is admitting she is petulant and childish enough to want to tie Romeo to her like a cruel little spoiled girl.
Juliet's use of the phrase "wanton's bird" comes in Act II Scene II of Shakespeare's Romeo & Juliet, which is the very famous balcony scene. Romeo has climbed the outer walls of Juliet's home to catch a glimpse of her and finds her on the balcony pouring out her love for him. When Romeo hears this, he is overcome with emotion and responds to her. During their exchange, Juliet realizes it will mean death for Romeo if he is caught, so she tries to urge him to go.
"'Tis almost morning. I would have thee gone.And yet no further than a wanton’s bird,That lets it hop a little from his handLike a poor prisoner in his twisted gyves,And with a silken thread plucks it back again,So loving-jealous of his liberty."
Dude. You need to get your head checked. Who reads this stuff?