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.