He killed himself over his mother finding out that he slept with a girl?

Haven't you ever heard of "mother issues" before? Poor Billy was obviously brought up by a domineering mother, who happens to be best buddies with Ratched of all people, and probably instilled a strong sense of guilt in him from a very young age.

Deep seeded feelings of Shame, Guilt and Anxiety have a power of their own and can make you do strange things knowingly or unknowingly.

Uh there was a reason why the kid was in the institution, and Ratched knew how to set him off. She just couldn't take it that McMurphy had cured him of his lack of confidence and stutter so she had to knock him down by threatening to tell his mother.

"I really wish Gia and Claire had became Tanner" - Honeybeefine

I never wondered why he killed himself in the first place.

He's crazy dude.