Why did she care so much? She's supposed to be a cold-blooded assassin of myriad victims. The star of the dark guild. Yet she behaved like a school girl here, when her towel fell. Yuuya is so broken psychologically, due to being a chronic bullying victim both at school and home, that it's no wonder he acted like he had accidentally broken someone's priceless Ming vase, but she should have remained largely indifferent. She ought to be quite dead inside, with suppressed emotions, considering her profession.
She's head over heels in love with him, of course. She may be a hardened badass killer, but deep down she's still a human being, with all the weaknesses and basic desires as the rest of us. It's already been established that due to his looks and demeanor, Yuuya is quite the disarming fellow to anyone who doesn't already have enmity toward him, and even more so to the opposite sex. Luna has arguably spent more time alone with him in their training than anyone else, and this after he literally saved her life with no expectation of recompense, so it's hardly surprising that she'd be so smitten with him.
Not only that, but she clearly has a poor opinion of herself (arguably justified, given her occupation), and a (well-justified) high opinion of Yuuya. She's already feeling shame just being in his presence. Add in the nudity of a body she's been hiding nearly since birth just to survive, and it's hardly a surprise that she wouldn't want him to see her naked.