Trouble understanding the notion of interconnectedness I started meditating regularly a few months agowith the aim of practising mindfulness,  primarily using Headspace.  I also read Sam Harris' Waking Up and listened to the Science of Mindfulness lectures by Ronald D. Siegel. 

I find a lot of the ideas very compelling (living outside of the present moment as a source of unhappiness, the impermanence of all things) and I am also beginning to see the benefit of meditation for my mental state.

The only concept I have trouble taking on board is the interconnectedness of all beings.  I have read various descriptions, some coming from the modern, Western take on mindfulness and this notion still doesn't resonate with me. My reaction remains 'we're not connected whether we believe in separate selves or not.'  So perhaps I'm taking this too literally. Is interconnectedness just a way of saying 'we're all in the same boat and will have a run of bad luck in the end' rather than some sort of mystic connection?

I'm not sure how clearly I stated what my confusion is but I would appreciate any insight.
"Interconnectedness" in Buddhism arises from the principle of "dependent origination". Dependent origination is an alternate way of viewing cause-and-effect relationships. Rather than saying "A causes B", (that is, one fixed object or idea leads to another fixed object or idea), it says instead:

"B arises and is supported in dependence on A, so that if A passes away, B also passes away". 

That then extends to the fact that A also depends on supporting conditions, and those supporting conditions also depend on other supporting conditions, and so on and so on ad infinitum. Nothing exists independent of supporting conditions that allow it to continue existing.

So your "being" exists dependent not only on other beings, but on a whole mess of other conditions (such as the composition of your body, the environment...) and all those things also have infinitely long chains of dependence. As a result, everything is in flux and nothing has a completely self-referential, independent existence.

(This, by the way, is the source of a common misconception that Buddhism teaches that "nothing exists" or "nothing is real". It does teach that nothing exists absolutely, but things most definitely do exist in relation to their supporting conditions, which is why you should still look both ways before crossing the street :) )

You can take this on a very empirical and non-mystical level. But through practices like vipassana where you start to get a felt sense of the constant flux, the constant creation/destruction of your own experience rooted in ever-changing conditions, welllll... you could start to feel that there's something mystical (or at least, indescribably big!) about it.

Editing to add something more to the point of your question: Practice also brings ever-widening insights into the way that you impact others and the way they impact you, which aren't necessarily "mystical", but they are subtle, and call into question the boundaries of your personality. It's more mundane than "I am one with the universe"; but it's richer than "stuff happens". Somewhere between the two :)