Answer:

The question states which quote represents her independence let's analyze each quote.

A)“Habitually obedient to John, I came up to his chair: he spent some three minutes in thrusting out his tongue at me as far as he could without damaging the roots…”    This quote is incorrect because it says she is obedient to John.

B)“You ought to be aware, Miss, that you are under obligations to Mrs. Reed: she keeps you: if she were to turn you off, you would have to go to the poorhouse.”    This quote is incorrect because it says that she is under obligations of Mrs. Reed.

C) “What do I want? A new place, in a new house, amongst new faces, under new circumstances…”     This is correct because she is talking about what she wants.

D)I had had no communication by letter or message with the outer world: school-rules, school-duties, school-habits and notions, and voices, and faces, and phrases, and costumes, and preferences, and antipathies—such was what I knew of existence.     This is incorrect because it says she has no communication with the outside world therefore not independent.

In conclusion the answer is C