Hi Annker
Welcome to the Forum. What you describe, sounds typical hedgehog – they like to be their own hog! Some hogs won’t use a hog box, until it becomes part of the landscape, so one might use it in time.
I wouldn’t feed eggs, cooked or raw, personally. Hedgehogs are known to eat eggs occasionally in the wild, but these would be much smaller than chicken’s eggs. I think you are better off sticking to cat/dog/hedgehog food.
You might need to create a cat proof area if they are a problem. If I was starting again, I think I would use a piece of perspex type material on bricks so that it is high enough for the hogs but low enough to deter the cats. The reason I say that is because I have found over the years, that some hogs prefer to eat outside and because there are openings on all sides, that system seems to work. You will, of course, need to put bricks on the top of the support bricks, as well, to stop the cat lifting it up. That way you still have very good visibility. It is a bit of an ongoing problem and if you read some of the old posts you will find all sorts of other solutions.
Good luck.
