External cameras for diagnosing aircraft issues are not used on any current aircraft. As mentioned in another answer and question, some aircraft do have external cameras, but these are more for maneuvering on the ground.
One reason for this is that cheap, high definition cameras that can stream directly to a monitor in the cockpit are a fairly recent invention. Previously, such an addition would have been fairly heavy and expensive to add to an aircraft. As mentioned, some newer aircraft that have digital cockpit displays do have external cameras for other purposes. However, even in aircraft with cameras, they are generally not used for diagnosing issues.
A lot of aircraft issues, especially with engines, are not readily visible from the outside. Internally placed sensors are still much better at detecting issues. Landing gear in particular may appear to be extended but not locked properly. Pilots are more concerned with how functional something is. Modern aircraft are equipped with hundreds of sensors and these are designed to give the pilots all the information they need about the functionality of the aircraft. These sensors are much more simple and precise than a camera. External cameras may give a misleading picture of the aircraft's state, and decisions lean towards an abundance of caution. Unless there is some extremely obvious discrepancy, it's unlikely that a camera will be trusted over the sensors.
Of course there are still incidents such as fuel leaks and fires that external cameras would help to identify. However, the aircraft sensors should be able to detect these issues once they are serious enough. Pilots also get information from passengers/crew in the cabin, and from external observers in other aircraft or air traffic control. Cameras are also limited in where they can be placed to provide a good view of things, and how well they will perform at night or in bad weather.
There may be a general case for cameras being helpful in identifying issues, but these situations are rare enough that is is hard to justify the expense. If something is enough of a risk that you need to point a camera at it, you're better off just fixing it so it's less of a risk.