Why won't June marry Neil?

At the beginning of Chapter Six, Lily meets Neil for the first time. Upon meeting Lily, Neil asks a series of questions that Lily feels she will not be able to avoid, such as where she is from, how long she is staying, and so on. Neil is in love with June and has continually asked her to marry him. June keeps refusing; she had been engaged before, but her fiancé never showed up to the wedding. When the topic is brought up in front of May, she begins to sing “Oh Susannah” and has to leave for the wailing wall.