What kind of bike is best for bike camping?

That depends on what type of riding you want to do and how far/how fast you want to go. Any mountain bike will take you on dirt roads and single track, but true all-terrain bikes (with thicker, knobbier tires) ride better over everything from rocks to sand to the pavement. Road bikes are wonderful for multi-day, self-contained tours on gravel roads and paved country roads. A touring bike with drop bars is best for easy riding over long distances if you are doing a section of the TransAm or something similar.