The pilot reported that, following landing gear retraction after takeoff, the landing gear unsafe light remained on. He reselected the gear down and obtained three green lights but the unsafe light again remained illuminated when the gear was selected up again. Suspecting that the problem was related to a microswitch, the pilot continued the flight. However, the nose gear collapsed during the landing roll. The nose gear drag brace strut connected to the landing gear strut had failed, allowing the gear leg to hang free. There was evidence that the part had been cracked for some time until it failed on gear retraction on this flight. The initial crack was induced by excessive forward load on the gear leg. However, the actual source of this load was not established. No specific inspection of that part of the aircraft was required, nor was the area easy to see during maintenance or preflight inspections. This accident was not the subject of an on-site investigation.