I agree with most of Robin's argument, but does it really answer Bryan's questions? I feel both of them are right, although in different ways.
The tricky thing here is the subjective feeling of continuity of self. We experience a break of continuity when we fall asleep in a normal way:
I lay me down and slumber
And every morn revive.
Whose is the night-long breathing
That keeps a man alive?
A.E. Housman (1859 - 1936)
and also when we die in a normal way; because continuity of self is a feeling, it is subject to reaction times, so arguably a person who happens to be in the epicenter of a nuclear explosion won't notice anything, no subjective break of continuity will take place — their brain will just cease to exist. I think this is the key to the whole argument: if you stop a person's mind so fast that the mind doesn't have time to notice the fact that it's stopping, then both the mind (after it is restarted, possibly in a different medium) and any external observers can subjectively agree that continuity has been preserved. I understand that the current cryonics implementations don't work this way, which is what Bryan argues.
However, is continuity such a big deal? After all, we are all used to breaks of self-continuity, even if we don't usually regard them as such. When I wake up every day, "I" am a different subjective continuity, and it is by sheer force of habit and by the fact that the new "I" has rather detailed "inside" memories of my previous continuities do I identify with them. Suppose a person falls asleep normally and, while asleep, dies of monoxide poisoning or heart failure — the person's mind will never be aware of this, because its sense of self-continuity is not operational. It just ceases to exist. If it can be somehow restarted, how is this different from waking up? Essentially, a new person (self-continuity) will wake up and say, "Where's my coffee?"
Also, if I woke up without any memories of "my" previous continuities (supposing further that such memories cannot be restored), what would it mean to say that I am me, as commenter Blackadder says in the other thread? In a trivial sense, I would be me because I definitely wouldn't be you or any other person, but that's about as far as it goes.