Eggnog most definitely DOES have booze in it!! I remember my mom making it every Christmas...usually with brandy or rum....along with eggs,cream and nutmeg.
There is alcoholic and non alcoholic versions of eggnog. The non alcoholic version tasted like if you mixed milk, with a pinch of cinnamon and a pinch of nutmeg and a splash of non alcoholic advocaat, if there was such a thing!