@John: The structure recovers its shape after compression is released, this is why I'm considering it as hyperelastic. I updated the material model adding the tensile testing data, the best fit was with 5 parameters-Mooney-Rivlin model (Ogden models did not fit properly), but still experiencing the same issues.
In any case, the structure does not bend on itself during simulations, so there are no self contacts at all.
@Bill: What I'm trying to simulate is the compression of the whole structure which has a 10x10 mm base dimension. The picture above show an element with edge of 1/14th of the entire structure (i.e., 0.714 mm). If we talk about single "column" then yes, you are right, it is one half of its edge (1/4th considering the cross-sectional area). In any case, both simulating the behaviour of a single column with or without simmetries still show no buckling.
Also, switching to SOLID187 to SOLID285 elements didn't improve the solution.