Juliet feels thwarted initially. When she says "wherefore art thou Romeo," it is as if she is railing against fate for making her fall for an enemy. After that, she feels the way most of us feel when we realize life is unfair: angry, frustrated, and resistant. If family is the problem, then she rejects her family obligations, telling Romeo to "Deny thy father and refuse thy name;" or, failing that, offering to do the same for him. I don't think she's thought this through at this point. She is very much like a teenager who sees black and white, and hasn't thought about the implications of what she calls for. So her impulse is to reject their identies. This shows how deeply troubled she is.
She knows that the Romeo she has fallen for has none of the markings of an enemy, certainly not any of his body parts. So she has two options: change reality "O be some other name!" or argue herself into rejecting the possibility that the family feud could extend to him. So she experiences conflict between doing right by her family and having her heart's desire.
And she wants her way.
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