OK - I am going to make the case right now that it is better to choose both boxes:
I like your setup there, Einfach. And you make a pretty solid case - taking both boxes does, in fact, guarantee that you'll get at least $1000. You correctly point out that by taking only the closed box, it is still possible to receive nothing.
But let's bring in some decision theory to this. Right now, you're basing your decision on the fact that you're at least guaranteed some money. Nothing wrong with that stance, but the proponent of a decision theory based assessment is going to say you should maximize the utility of the situation. In other words, you shouldn't be satisfied with receiving a guaranteed $1K. Instead, you should want to maximize the amount of money you can make.
The assessment is going to be based on the confidence of a particular outcome. In your case, given the parameters of the experiment, it seems okay to say that your confidence of getting at least $1000 is 1 (that is, 100% confident). The outcome space for this event is 2 (getting $1K and getting an additional million) and is fully exhaustive.
But what is your confidence that you'll receive $1,000 versus the confidence that you'll receive $1,001,000? Let's say you have a confidence level of .95 that the being will accurately predict your decision. Thus con($1000) = .95 In other words, the confidence you have in getting $1000 is 95%, and the confidence in getting the extra million is 5%.
Note this is far more conservative than the problem is suggesting, which seems to be a .999 confidence level. I just think that .999 is a little high, and the argument I'm using still works for .95
So, you have .95 confidence that you'll get $1000.
But if you decide to take the closed box (and again, assuming you are 95% certain the being's prediction will be accurate) then you get: con($1,000,000) = .95.
Now, the decision theorist has a way of calculating the "monetary value" of a particular outcome by applying the confidence level to the monetary gain. Since choosing both nets you at least a grand, your monetary value will simply be $,1000 for that event. You're not expecting anything more (nor should you be) and you're guaranteed at least that amount.
For both boxes, the monetary value would be something like $950,000 given your confidence level. You are almost certain to receive a million smackers by choosing the closed box.
The decision theorist wants to argue here that the preferable bet is the one with the greater monetary value - taking the closed box. Otherwise, he would argue, your decision isn't rational because it fails to meet the Kolmogorov axioms of probability (thus leaving you open to Dutch Book bets).
Put simply: isn't it worth taking a slight risk to maximize your utility over getting a sure, but much smaller, amount?