If that's what the doctor told Becky, and Becky's first thought wasn't to get a job and get the fuck out of there (because she is not under any legal obligation to do any of that nasty shit for Amber), then Becky is an absolute moron and can go cry to someone else, lol.
And they didn't break up, Amber just used her special Amber grammar which made it sound that way. Allegedly.
Still Becky is a sped, plain and simple, she likely didn't think of this in terms of freedom, but in terms of 'I have to take care of her'.
Do you think if the doctor really did tell her Hamber was dying, she'd cry though?