Just how big is your ice box drain?
Unless your ice box is very unusual, it is not hard. Most ice box or refer have a very small tube for drain. They sure don't need any chemicals to clear it, nor a snake. Usually just a long wire, like a coax cable for your TV. Coax cable is great because it is somewhat flexible and would bend slightly, and it is coated.Don't use a straightened coat hanger, since it may be sharp and poke a hole on the side of the tubing.