It seems to me that you are assuming that I was/am a clingy guy and that she is breaking up because she is bored of it.
I'm not making that assumption
You (and I) don't know the reason why she is being distant, right? We only know that she is has not cut you off and she has not invited you in. It's limbo now. She flutters in and flutters out at her convenience.
This is the hardest thing for men to deal with. You don't want to do anything to push her away and at the same time you don't want to look like some worthless sap who will jump for breadcrumbs. There is very little you can do in this situation to improve things and a lot you can do to make it worse.
Ideally, when she opens up a bit more, it would be good to have lunch and ask her what is wrong and listen. Then think - maybe chat here - and then decide what to do.
In the mean time, just be very nonchalant for a few weeks, if she doesn't pull more on you, you could ask for lunch. If she says no, I would start thing in terms of exploring other options.
Does that help?