It wasn't so much Hinamori's arguments as her emotional state that Hitsugaya paid attention to. He preferred to avoid upsetting her whenever possible.

He did have sick leave, and no, he wouldn't lose his job if he called in. It was more the principle of the thing; stopping what he normally did felt like he was losing to the person that did this to him. He didn't like losing.

He was about to give in to Hinamori's suggestion that she accompany him -- for her peace of mind, not for his own safety, thank you -- when Kuchiki decided to bring up an unsettling possibility.

"... I'll call in."

Later, he would be annoyed with Kuchiki's tactics here, but right now, all he could consider was Hinamori's safety.