Depending on the Tanna, Koheles and Shir HaShirim might not be considered a part of Tanach, but merely written from Shlomo HaMelech's wisdom. Therefore, unlike books of Tanach, one who touches these scrolls does not become tamei (see Eduyos 5:3, Yadayim 3:5, with Peirush HaMishnayos to Eduyos ibid.).
Does this mean that, according to those opinions, one may even carry those scrolls into a bathroom, or speak verses of them, unlike books of Tanach? Or are they still considered Torah and still cannot be brought into a bathroom, but because they were not composed with Ruach HaKodesh, they weren't able to be canonized into Tanach?
(Just to clarify, we pasken like Beis Hillel, and therefore these books are a part of Tanach halacha l'ma'aseh. Therefore, I have not tagged this question halacha.)