Day had thrown a knife at Matias just outside of the hospital he broke into. Since Matias died, June wanted to seek revenge on the person who killed Matias--who she thought was Day at the time.
Involuntarily, June was chosen to participate in a Skiz fight and had gotten a stab wound mid-battle. After an attempt at escaping, the furious fight betters attacked her for retreating. That was when Day--who was watching the fight on a building ledge--decided to save June. Within a couple of days, June healed and she even fell for the guy who saved her--not knowing that he was Day.
Day pretended to die so he could care for his family so he often visits his home whenever he has items for them.