I guess just something that will make a thick consistency. Maybe use water (sparingly) instead if the egg whites? There are pieces of gingerbread that don't have icing on it on my house, but there's no lollies on those bits either (the lollies stick on with the egg whites/icing mixture).

If you can't get the icing to work with anything other than the egg whites, there is an edible bakers glue that you could get from baking supply shops. You could use the icing to put the house together and then the glue for the lollies.