How to remove this frustating and strange echo?

Well, at least I can hear it now...

But that's the good news - the not-so-good news is that there's no way you are going to fix this - it's not just an echo, but is actually delayed feedback. This is where sound from one party is played on another party's machine and picked up by their microphone, and fed back to where it came from, and then it goes round again - with a delay at each stage, which is why it appears to repeat. In meetings this usually happens because somebody can't be bothered (or hasn't got) a headset - which effectively prevents this from happening. In this instance the sound is somewhat modified by going through a lossy transmission system, which is why it sounds more like squeaks.

The reason that you can't remove it automatically is that the frequencies this is happening at are also the ones you want to keep; it occurs in the speech band, because it's speech that starts it off in the first place. And there's no mechanism that can 'hear' just the speech - that's a human ability that machinery doesn't have. In principle there are potentially other solutions - if, for instance all the squeaks were at a level that was lower than the speech, then they could be at least limited by level-gating. This is where you get rid of all the sound below a threshold level that you set. My experience of this is that it's really difficult to set a sensible threshold, and then a couple of things happen; firstly all the loud bits are still there, and secondly, quieter parts of what you want to retain are lost anyway, because of the gating.

So the problem is that machines cannot understand speech in the same way that humans to - and even if they could, it's going to take a pretty expensive bit of kit to sort this sort of mess out. So your only hope here is to recognise what's actually causing this, and try to attend to it at source if you need to do it again.

I'm sorry we can't be any more help than that. This situation occurs quite frequently, but there's nothing that can realistically be done about it at present.