ethereal, I think your instincts are right -- you are experiencing detachment and indifference, not non-attachment. Speaking from what I know (Buddhist perspective), wanting to be free of unpleasant experiences is a form desire and ego craving. Both the desire to have and aversion - the desire not to have - are forms of attachment, as you suggested. Equanimity is described as the mental quality of inner balance and strength that counters ego craving and aversion, and there are various techniques to develop it. One important thing to note is that equanimity has two opposites - the far enemy and the near enemy - which detract from your ability to sustain it. Anxiety is the far enemy of equanimity, and we can clearly see how anxiety is the enemy of equanimity. But the near enemy is something that we could easily mistake for the quality itself, and in this case, it's indifference, which seems to be what you are experiencing.
One way of developing equanimity is to meditate on impermanence. We can never permanently escape from unpleasant experiences, no matter how much we wish we could. Our lives are always in motion between up and down, gain and loss, satisfaction and disappointment, and no set of circumstances, good or bad, lasts forever. Sometimes we might feel we are the only ones in the whole world who experience that, but if you look closely, you can see that this is a basic fact of existence that everyone experiences. Equanimity means that sometimes the world is not the way we want it to be, and sometimes it is. This is ok. Some might say that accepting the way the world is means that we are not motivated to make changes, but how can we make changes if we have obsessions? We can't clearly see what would be most beneficial and effective, and then we are discouraged and depressed and give up.
Some might say that in very difficult situations, its impossible for anyone to sustain equanimity. What if someone close to you died suddenly? In fact, it is possible to do it, it's just very, very hard, even for people who are advanced on a spiritual path. The good news is that it is not necessary to hold ourselves to such a high standard. I would say that even equanimity with our goals and desires is too hard to begin with. Can we sustain equanimity when someone fails to return our phone call promptly? What about when the line at the supermarket is slow? Or if we disappoint ourselves by forgetting to do something important? There are millions of insignificant events that we react to throughout the day that disturb our equanimity, and these should not be overlooked! They are important because mastering them gives us the sense of how to balance ourselves with an uncertain world, and gives us an intuitive feel for how to approach bigger things. Wrapping ourselves in a soft cocoon of indifference where we don't feel anything is unhelpful, because its only by noticing in great detail what lack of balance feels like that we can learn to be balanced. From the point of view of cultivating equanimity, its not very useful to say to youself, "Oh no! I'm unbalanced! I must escape from this terrible feeling!!" Its better to simply notice the feeling, knowing that it will eventually pass and extend compassion toward yourself by remembering that everyone feels this way sometimes. Of course, there's no point sticking around trying to cultivate equanimity in a truly overwhelming situation. If its too hard, try something a bit less challenging.
I found
this talk on equanimity very helpful. Perhaps other will too.