We die according to our personality. We deal with our approaching death in the same way we have handled other challenges in our life. Our personality doesn’t change it becomes stronger, more defined in the ways we have addressed living. Think about your brother's personality and the manner in which he has lived his life. See if his reaction to his life threatening illness is the way he dealt with his other challenges. His response seems very dogmatic, almost a form of denial, also unemotional. It certainly is a way of keeping people distant.
For some people they just don’t want to be constantly reminded of their frailties. They don’t want people to tell them what to do to get better (when most don’t know what they are talking about anyway) and they don’t want to hear how good they look (when they know they don’t). Often illness and how “we are” becomes the only topic of conversation. Your brother may be wanting to avoid that from happening.
What to do? I guess you have two choices. Honor his wishes and play the game of not talking about this life challenge or have a conversation with him as sibling to sibling. Tell him your feelings and your need to have an open talk with him. Tell him how other family members are reacting. Ask if he can help all of you deal with his life situation. If he didn’t know his life was ending and there was a family occurrence would everyone ignore it or would they come together to support each other?
We often think because a person is facing the end of their life we have to do whatever they want us to, regardless of how it affects us. We treat them as we treat a very spoiled child--do what they want so no one is uncomfortable all the while everyone is uncomfortable. I think it is perfectly acceptable to have an open, honest talk with your brother about his approaching end of life. Have the conversation and then come to a compromise so that both of you (and the rest of the family) can find a comfort zone that satisfies everyone.
Something more... about "I Don't Want To Talk About It"...
Fear is powerful. Perhaps reading A TIME TO LIVE, Living With A Life-Threatening Illness, your brother could gather some tools to help him with his gift of time. He would better understand what was happening and reduce his fear. To understand the dying process read GONE FROM MY SIGHT, The Dying Process. Both booklets are in the End Of Life Guideline Series bundle.