Just wondering if it could be mealy bug though I've not seen it on box. - it doesn't look like powdery mildew. Look closely at the white stuff and see if some is just fluffy (eggs) and other parts are actually a white insect. If in doubt try squashing - if its a mealy bug it will stain red . They are easily controlled by painting with methylated spirit using eg a watercolour brush.

Don't think it's mildew, I've got it on my box as well, not sure what it is but it is sticky have not seen any creatures.

it looks like either wooly aphid or mealy bug hiding in the white fluff. pull a but off and tease it apart. if there is a small critter in it then that is what you have.
you can get sprays for both pests.

The creatures are not easy to see until you know what you're looking for. Try the meths first, its cheaper than bought insecticide. (Though it will take longer to apply than spraying...)

Looks like Box Aphid. Give it a shake and you should get a small white cloud. Most systemic bug killers should deal with it.