The paper considers convergence, accuracy and efficiency of a block J-Jacobi method. The method is a proper BLAS 3 generalization of the known method of Veselić for computing the hyperbolic singular value decomposition of rectangular matrices. At each step, the proposed algorithm diagonalizes the block-pivot submatrix. The convergence is proved for cyclic strategies which are weakly equivalent to the row-cyclic strategy. The relative accuracy is proved under the standard conditions. Numerical tests show improved performance with respect to the block-oriented generalization of the original method of Veselić. Combined with the Hermitian indefinite factorization, the proposed method becomes accurate and efficient eigensolver for Hermitian indefinite matrices.