Do these types of nurses require specific degrees?

Degree doesn't matter: for the most part, a diploma holder, an ADN, a BSN or a MSN can all be nurses in the specialties listed above. All they need is to have passed the NCLEX and hold a valid RN license. In the US, any of those four education options above will make you eligible to sit for the NCLEX, and the license you get is the exact same for any of the four.

That being said...some employers may require that the nurse hold a specific degree, such as a hospital choosing to hire BSNs only for their oncology department. Doesn't mean that the diploma nurse can't ever be a oncology nurse--just maybe not at that particular hospital. She/he may find that job elsewhere.

Also, some states may impose requirements on certain specialities. For example, California requires that public health nurses hold BSNs or higher.

As the previous poster mentioned, those types of nurses can certainly enter their respective specialties with a diploma, ADN, BSN, or entry-level MSN.

Some facilities might demand experience or specialty certifications prior to hiring. However, one generally does not need a specific degree to break into the aforementioned types of nursing. Due to the economic situation, some hospitals have resorted to a preference for nurses educated at the BSN level.