Will it spoil bc of the egg whites?

The absolute easiest way to glue together a gingerbread house is to melt gummy candy - bears, worms, you name it. But to answer your question, this recipe is a version of royal icing. That stuff hangs around for years, and although it isn't very tasty, it is edible months after the making. Sugar is one of the original forms of preservative, after all. I made a very large gingerbread house for a competition once upon a time, and kept it for five years afterward. The only reason I gave it up then was because we were moving and it is really, really hard to dust a gingerbread mansion anyway. I wouldn't have recommended eating it, but yeah, it was pretty much demolition proof. When my kids were younger, it took the meat mallet on New Year's Day to reduce a house to parts. And really, only the candy was "edible" by then.

The egg whites bind and harden with the sugar to form a decorative cement.
Although technically edible, it is very porous and will absorb moisture, any airborne scents and germs so probably best to not eat it after it has been sitting out for awhile.
I find it easier to glue the house pieces together with edible glue (easy to make with corn syrup) and more economical to use meringue powder rather than fresh egg whites for the decorative Royal icing.
If you do use the icing as cement for the gingerbread structure, you will need to allow for the icing to dry on the walls before putting your roof on to avoid it collapsing into a festive mess…Also if planning this as a kid’s activity, kids don’t want to wait a few hours/overnight to dry the walls before putting on the roof. In my experience, they just want to get to the candy part!