Headshot photography is an art, not a science. Too many aspiring headshot photographers and even some professionals view headshot photography as a science, believing that all headshots should be taken exactly the same. Unfortunately, headshot photography is nowhere near that simple. There is no "one technique fits all" solution. This especially applies to lighting for headshots.
In general, it is great to use soft light, however certain occasions definitely call for hard light. I will attempt to discuss which situations require which light, but again, remember that there is no "one solution fits all" for headshot photography. Therefore, the below ideas should be taken with a grain of salt.
Happy and feel-good photos: These should usually use soft lighting, with few shadows. The only reason to really use shadow in these shots is for sculpting the face and showing what you look like. If you don't have any shadows at all, there won't be much interest in the shot and it will look flat and boring. On the other hand, too much shadow will hide portions of the face and could be distracting.
Dark and serious photos: These need a bit more shadow to add drama to the photo. But remember, the lighting should still be fairly soft because you don't want any distracting elements, and hard lighting can sometimes be distracting. Additionally, it helps to carve out features in the face more which can lead to more pronounced blemishes and such.
The easiest way to have more control over lighting for headshots outdoors is by using a bounce board or a flash unit. But remember that when using a flash, you can't shoot as quickly. I opt for a bounce board while paying an assistant to help out on the shoot.