Gaku being blunt probably wouldn't put Everett off, just more make him speak to him in a similar fashion. He would be factual with him, in hopes Gaku would appreciate it and be less ... unwelcoming, I suppose is a good word for it. 
Everett probably wouldn't know Eli very well, as she is new, but would have seen her around and has undoubtly worried about the girl's health, but that would be beacuse he'd never would have seen her in his care. 
He would know Alex, and would be on good terms with her because she's kind and he can sympathize with her lack of social skills.
Henry would be a familiar face, not only from the clinic but also from childhood. They would speak comfortably. 

I was in a roleplay on another site and part of the CS was a relationship chart. Should we do something like that?