OK........a lot been written while I have been away..........
In our instruction sheets for polystyrene hives, devised through our own experience laid on top of the advice from Scandinavia (sometimes incorrect btw) we recommend the following:=
Paint all parts of the hive which are normally exposed to the light. This means all outside surfaces. It is NOT necessary to paint the internal surfaces of the boxes, floor, or roofs, as the bees do this themselves with an imperceptibly thin layer of propolis. they varnish them, and when the box is fairly new you only see it where they fill in the tiny gaps between the beads of material. the rest still looks white, but set it against a split new one and you will see the slight difference. They do this to wooden ones as well btw.
So.....paint all outside facing surfaces. We use exterior gloss, but acryllic emulsion(water based) masonry paint is also satisfactory, and is the best paint for feeder insides.
Gloss paint, thinned as directed to a consistency recommended fopr spraying actually bonds with the box surface as it dissolves and mixes with a microscopically thin layer of the surface. Unless you go OTT witht he thinners yopu will never notice that this has occurred, but done correctly the paint never ever flakes.
So....Outside surfaces gloss or masonry paint.......inside the feeder (syrup compartment only) masonry paint.
Undersides of floor need not be painted as the algae do not get enough light there to grow, and so that surface stays just fine without paint.
All surfaces that the bees have access to inside the hive (interiors of floor, all boxes, and underside of feeder) need not be painted.
Only variant to that, and may be where the confusion arises, is that the material is rather translucent, especially in areas where it is thin. Our recommendation was for our National boxes, and ONLY our National boxes, where to ensure full compatibility with wood the front and back edges at the rebate are rather thin, that to prevent the bees trying to chew out to the light you should paint that area with a coat or two of polyurethane varnish. This puts a tough surface on the material and even the most persistent chewers cannot get into it. A secondaruy benefit is that it renders the surfaces more easily scrapeable.
As an aside, in Denmark, even with high density materials, I observed then painting the bottom edges of the front and rear of the boxes with this varnish, and the same reason was given, prevention of chewing. I am presuming it to be a precautionary move, with a lot due to past experience with 70g/L and 80g/L materials. The Nakka ones, in the photo earlier in the thread of the big stack of poly boxes, were normally made at 80g/L, and yes, I have lots of them, with very little or no attrition rate since buying them 11 years ago. Not anywhere near as good as the one piece versions at 100g/L and more that are the normal today. (This preceding paragraph relates to Langstroth btw).