I would announce a serious illness on social media because it's a wonderful way to get support. In addition, it's less painful to have to announce it once than to meet individually with each friend and family member to tell them the bad news. Each person deals with illness differently, but I've seen many people get wonderful support when sharing bad news on social media.
It's hard to put into words how profoundly social media platforms are impacting health for the greater good. Patients can reduce the impact of isolation and directly connect with others not only for friendship but to also share the latest treatment information, learn ways to mitigate side effects, gain wisdom from other countries and cultures, and more. In other words, we can take better care of our selves while dealing with a life-threatening condition because we can now build a whole "village" of support.
People are going to far today with this social media epidemic. Most people feel the need to share every intimate detail of their lives on their social media just to get attention. For this reason, I would not announce an illness I have on social media. However, if the purpose of announcing it would be to ask for prayer, bring awareness to, etc., then announcing it would be more acceptable.
I would not announce my illness on social media. I would talk to each of my loved ones individually and tell them all in a way that fits the person I am talking to. It would be insulting for some to hear about it through a secondary medium, the personal touch is often better.