Who is leaving? Why must she keep it a secret?

The answer to this question can be found near the end of chapter 19. The first half of this chapter contains narration about Lyddie attending one of Diana's meetings, and Lyddie leaves frustrated that she missed her chance to sign the petition. Diana offers to walk Lyddie home, and Diana laments at one point that she will miss Lyddie. Lyddie doesn't understand because Lyddie knows that she herself doesn't intend on going anywhere. Diana clarifies by saying that she will be the one leaving. She has plans to go to Boston. Diana is leaving so that she won't be dismissed from the factory for being pregnant. Diana isn't married, so the news already a little bit scandalous; however, it is made more scandalous by the fact that the father is already a married man. The entire thing has to be kept a secret to protect her reputation, the father's reputation, and not bring dishonor to the Association.

"I'll need to go soon. I can't bring dishonor on the Association. Any whisper of this, and our enemies will dance like dervishes with delight. . . I won't hand them a weapon to destroy us. Not if I can possibly help it."

In chapter 19 of Lyddie Diana Goss has to leave the factory, much to Lyddie's great sadness. Diana has been an inspirational figure to Lyddie, as she has been to all the girls working at the factory. She is an outspoken activist who is passionate and forthright in arguing for better conditions. More than anything else, though, she has been a real friend and mentor to Lyddie. So it's not surprising that Lyddie should be feeling so down at Diana's imminent departure.
Lyddie notices that Diana doesn't appear too well, but she's not quite sure what's wrong with her. It turns out that Diana is in fact pregnant, and since pregnant women aren't allowed to work in the factory, she has no choice but to leave. Diana needs to keep her pregnancy a secret because her unborn child has been conceived out of wedlock. In those days there was a great social stigma attached to illegitimate children and their mothers. The father of her child is a married man, a doctor, and so Diana's reputation would be ruined if the truth ever got out.