I'm guessing Nine saw her hold Yuuri against her chest, saying she will protect her, and assumed it was a declaration of love?
After being kept away from her for 4 days, and then kicked out right away to talk to Yuuri in private, it must feel like he's being forgotten/replaced. I can see why he'd cry.
...or maybe it's just jealousy that she held Yuuri, but pushed him away?