By universality, we usually mean not just the diagonal form; we mean a diagonal form proportional to the unit matrix. Universality is flavor-blindness.
It could be sufficient to have diagonal squark matrices – in the basis of superpartners of the energy eigenstates of quarks – for the unwanted new effects to be suppressed. However, it is extremely unlikely – unnatural – for both quark and squark matrices to be diagonal in the "same" basis. Such an accident would mean a fine-tuning that would require an extra explanation. Assuming we don't want to look for such an explanation, the only principle that may guarantee that the squark matrix is diagonal in a seemingly arbitrary basis is to assume that it is proportional to the unit matrix – it is universal, flavor-blind. This assumption is legitimate as far as naturalness goes because it increases a symmetry, the unitary rotation symmetry acting on the squarks.
The proportionality to the unit matrix is especially important for the first two generations. The third generation may fail to be completely universal and it causes smaller problems because the effects involving the third generation are suppressed due to the heavy top quark mass.