How is Jo rebellious? How does she rebel against gender norms? How is she "breaking the rules" about being a female in this time period?

Jo does not wish to be restricted or confined by the rules that society placed on women in that era. She felt that they did not allow her to express her individuality and be true to her own nature. She craved action and excitement that men experienced rather than the sedate role of a female. She was able to experience the action and excitement vicariously through her writing. Although she could pursue writing more easily than going off to war, even writing was not an acceptable occupation for a woman of her day.
To accomplish her writing goal she needed to move away for privacy. She also discovered that it was necessary for her to use a masculine pen name to even be considered for publication.
Her fiercely independent nature dictated that she must cut her hair and sell it for money rather than ask a relative for financial assistance. This choice reveals that she would rather look like a boy with cropped hair than be cast in a role that would cause her to appear needy or dependent.

Jo is rebellious against the typical female gender role of the Civil War era because she rejects most of the norms established for proper female conduct. For one, she resents her femaleness and longs to be able to leave home and do exciting things like go to war or live in the big city (she actually does achieve the latter after she rejects Laurie's marriage proposal). She is not romantic, and rebuffs Laurie's advances for quite a long time before he eventually asks her to marry him; again, going against the typical female role, Jo turns him down when almost everyone expects them to end up together. 
In order to get some distance, she moves away to pursue her dream of becoming a writer. Even this is made more difficult as a result of Jo's gender; she is initially rejected by editors because she is a woman, and it takes a great deal of time for her to be accepted as a female writer of serious works. 
Jo even has what sounds like a boy's nickname, and she prefers books to music, knitting, or other more properly female activities. When Marmee needs money, Jo's pride will not allow her to ask Aunt March, so she cuts all her hair and sells it! This is certainly not "proper" behavior for a girl, and she must live with her boyish haircut until it grows out.