That would be ''Bir kutu ayakkabı'' there is no such thing as ''ayakkabı kutu'' in Turkish.
When using a noun like box, and using another noun as an adjective because it is the contents of the box, the adjective is singular because the quantity is indefinite (like water or sand). I'm sorry, I'm not a linguist so I hope I used the correct terminology. Some examples:
Shoe box Coin purse Cookie jar Book shelf Sock drawer