After the runner has passed the first piece of Ice in my server during the paid abilities window prior to them approaching the next piece of ICE (Himitsu Bako) I use it's paid ability to return it to HQ. My understanding of what happens is that the runner remains in the same position on the server while the first piece of ICE will drop to where the Himitsu Bako was causing them to encounter it again should they continue the run. This has been the topic of discussion for quite a while in our local playgroup and was hoping to get some sort of official ruling on how it works.

That is a really thorny one. The FAQ states that when you pass a piece of ICE, you approach the next innermost ICE. Let me work this through.

You pass ICE a. Then in step 2.1 or 2.3, you remove Himitsu Bako from the server.

At the very least, that particular approach step is closed. Whatever happens next is a NEW approach.

We're either approaching the server, because we already passed ICE a, or we're approaching ICE a, because the server collapses and you approach the next innermost ICE. But we already did approach that ICE.

AHA. It's also on page 5 of the FAQ. "If a piece of ICE is destroyed during an approach or encounter with that piece of ICE, then the ICE is immediately PASSED". So no, you don't encounter ICE a again. You continue the run as if you had passed Himitsu-bako.