Why Didn't Red Kill Himself?

[Alardem]

Because it's "against the rules".

We see this with Eminess, whose attempt to quarantine himself and amputate his hand is overriden by the Tuurngait's desire to preserve his body. The only exceptions are those who manage to overcome the infection - Howard kills himself after communicating with the hive-mind, and Phillip can either himself at the end of Requiem or succumb to the infection.