For an internal connector to break requires a force to have been applied to it i.e. the external lead/plug to have been knocked in some way. No manufacturer would entertain a warranty claim for such a fault. What you describe is not a defect of material or workmanship but the result of user misuse at some time.
You have no argument to make against Fuji - just a decision to make about what happens next.
You may be able to get it fixed cheaper by an independent computer doctor but it will take longer and it's also possible that the defective connector has actually broken the board that it's fixed to - when it got knocked - hence the high repair cost quoted.
Good luck...
Peccavi
December 2009