Contrary to popular belief, there is nothing "healthier" about brown sugar than white. It's really just white sugar with added molasses, and that molasses is what gives it that softer, moister texture. When a recipe doesn't specify, use light brown sugar, which has a less pronounced molasses flavor.
You may also see granulated brown sugar in the supermarket. It's great for sprinkling on toast or oatmeal because it's dry and doesn't clump, but I don't recommend it for baking.
For more information on different types of sugars, see 7 Ways with Sugar.