By using the Saint Venant`s semi-inverse method stress-strain state of stretching nonlinearly elastic cylinder containing screw dislocation was analyzed. The ranges of material parameters when diagram of loading (the relationship between the axial load and the elongation of the cylinder) has a falling segment were defined. The existence of such segments can be treated as a stability loss of stretching cylinder. To analyze the stability the bifurcation approach was used that based on linearization of the equilibrium equations in the neighborhood of the obtained solutions. The bifurcation point was defined as such value of the "loading" parameter (Burgers vector magnitude, stretch ratio or other strain characteristic) for which the linearized problem has a nontrivial solution. Numerical determination of the bifurcation points was based on the analysis of the homogeneous linear boundary value problem of sixth order whose coefficients expressed through the radial displacement function and its derivative. The similar problem of compression was used for verification purposes.