Is brittle bone disease genetic?

Usually. Osteogenesis imperfecta is a disorder of brittle bones. There is an abnormality in collagen 1 function, and so you are not able to make bones or other connective tissue structures correctly. This is why people often have "blue sclera" because the collagen is abnormal. Most cases are genetic, but a new mutation (not inherited) can be seen (about 35%).
Yes. It is an inherited defect in the formation of one type of collagen that is essential for bone formation.