The questions you ask indicate that you do not understand the contents of the basic plate model (from which the shell model is derived), i.e., what normal stresses the model provides, and that you do not properly understand buckling analysis. The element type has nothing to do with the requirement to do a buckling check, and furthermore - as "FEA way" mentioned - nonlinear buckling analyses does not provide eigenvalues and eigenmodes, but rather requires some initial mode shape to estimate the imperfection that, when applied to iterative solution of the problem at hand, at some point produces a rapid change in the force-deflection curve.
I suggest that you consult your textbooks and colleagues for advice on how to proceed with dimensioning a steel column (strength and buckling) using non-linear analyses.
PS. The straightforward method to perform this analysis and dimensioning is to start with a beam model. Analyze and dimension it against bending and shear (which is usually uniaxial, and probably is in your case too), and then perform an eigenvalue calculation to determine the linear buckling load. Note that FEM is not necessarily needed for the aforementioned steps. If the beam passes both checks, it is probably adequately sized, but if you want to do more in-depth analysis, you may use the first buckling mode as an imperfection and then perform a non-linear analysis (increase load in steps and trace the force-deflection curve in the transverse and axial direction) to identify a non-linear limit load, which provides a more realistic (and always smaller than the eigenvalue solution!) buckling load.