Why does Betsy leave the mill in chapter 14?
Chapter 14 is mainly focused on Lyddie's increasingly selfish nature. She is the factory's best worker and consequently has been assigned the task of training a new worker. By the end of the first day of training, Lyddie is so frustrated with her pupil that she tells the boss that the new worker is ready for her own loom. Lyddie does this just to avoid the responsibility of additional training even though she knows her decision is dangerous.
Another event of chapter 14 is the petition being passed around for shorter work days. Betsy does sign it, which could cause her to get fired and black listed. Other girls that signed were let go and prevented from finding new jobs, but Betsy wasn't dismissed for her signature. Marsden says that Betsy is in poor health, and she was let go because of her cough.
Her cough is more worrisome than a cold or flu, though. As the above post mentioned, Betsy likely has contracted tuberculosis. Severe coughing could be a variety of things, but coughing up blood is a big indicator for TB. Tuberculosis is also contagious, so getting rid of Betsy protects the rest of the factory workers. At the time the novel takes place, tuberculosis is a death sentence for Betsy.
I assume you are referring to Betsy in the book Lyddie.
Betsy is sick with tuberculosis, or TB. She becomes increasingly sick, first stopping work, then being sent to the hospital. Eventually her uncle comes to pick her up and take her home. She claims that she will be back next year, but Lyddie knows she won't - she'll never be strong enough again to work in the mill.
The book doesn't specifically state that Betsy had tuberculosis, but the symptoms described point to what we now know as TB. Coughing up blood is a big symptom of TB.