Necessary to quit a HandlerThread?
My personal preference is to create a new thread whenever there is a need for it and clean it up when it's done. This way I don't have any problems with multiple components trying to use the same thread at the same time and I keep a "clean ship". Also Android has the nice AsyncTask which makes this easy for you.
That being said, I see no reason why you can't reuse your handlerthread for multiple components, provided you regulate access to the thread AND clean it up properly when your activities are destroyed. If I understand this post correctly, your thread may keep running even if all your activities are terminated because your process may keep on running. To solve this, you can set your thread as a daemon thread. Daemon threads are automatically destroyed when the last non-daemon thread in your application is finished.
BTW, alternatively you also might want to consider using a ThreadPoolExecutor