The substantive point here isn't about EU calculations per se. Running a full analysis of everything that might happen and doing an EU calculation on that basis is fine, and I don't think the OP disputes this.

The subtlety is about what numerical data can formally represent your full state of knowledge. The claim is that a mere probability of getting the $2 payout does not. It's the case that on the first use of a box, the probability of the payout given its colour is 0.45 regardless of the colour.

However, if you merely hold onto that probability, then if you put in a coin and so learn something about the boxes you can't update that probability to figure out what the probability of payout for the second attempt is. You need to go back and also remember whether the box is green or brown. The point of Jaynes and the A_p distribution is that it actually does screen off all other information. If you keep track of it you never need to worry about remembering the colour of the box, or the setup of the experiment. Just this "meta-distribution".