Yes Pro/MECHANICA could probably solve this problem. However, if your picture is right and the assembly has as much clearance as you show in the pictures it may be a little tricky.
The contact region tells mechanica what surfaces may come in contact during the analysis. That said this contact surfaces do not carry any tangential load (i.e. friction) only normal load (contact force). If the two surfaces start to slip between each other during the analysis(no Friction) then the analysis will fail. The work around will be to try to start the analysis in the position at which the surfaces are in contact or close to be in contact. Also sharp edges will crate singularities, therefore, a small round should be applied to the edge and the surface created from the round should be specified as s contact region. I hope this information helps
To extend further this discussion I want to ask though hydralulic jack is not meant for sustained load, but can we perform such buckling analysis for the hydraulic cylinder using contact regions in Mechanica.
In the case of the hydraulic jack you will have to assign constraints in either side of the piston and applied a force in the opposite side. The contact regions is only use to run analysis that require localize stresses in the contact area. Although, buckling analysis in mechanica is a linear buckling otherwise also known as a eigenbuckling analysis. This type of analysis uses the stress field to calculate the eigen values (the buckling loads) and the eigen vectors (buckling modes) of the structures due to the force in the model. The localize stress in the contact area should create an insignificant effect in the buckling analysis.
The word of caution when you using buckling analysis is that the results are conservative. Therefore, a safety factor of 2 to 4 should be used out of the results obtained from the analysis.