The h-extra connectivity and the h-extra diagnosability are key parameters for evaluating the reliability and fault-tolerance of the interconnection networks of the multiprocessor systems, and play an important role in designing and maintaining interconnection networks. Recently, various self-diagnostic models have emerged to assess the fault-tolerance in interconnection networks. These interconnection networks are typically expressed by a connected graph G(V,E). For a non-complete graph G and h≥0, the h-extra cut signifies a vertex subset R of G, whose removal results in G−R disconnected, with each remaining component containing at least h+1 vertices. And the h-extra connectivity of G is defined as the minimum cardinality of all h-extra cuts of G. The h-extra diagnosability for a graph G denotes the maximum number of detectable faulty vertices when focusing on these h-extra faulty sets only. The complete Josephus cube CJCn, a variant of Qn, exhibits superior properties compared to hypercube Qn, and also boasts higher connectivity. In this study, with the help of the exact value of the h-extra connectivity of CJCn, the explicit expression of h-extra diagnosability of CJCn under both the PMC model for n≥5 and 1≤h≤⌊n−32⌋ and the MM* model for n≥5 and 2≤h≤⌊n−32⌋ are identified to share the same value (h+1)n−(h−12)+1.