Super-cold LOX gives advantages in performance for a rocket. But it isn't used in all rockets. SpaceX's Merlin 1D rocket engine uses super-cold LOX. Also NK-33 use sub-cooled LOX. Could super-cold LOX be used in every rocket engine? Does it needs modification with new parts or does it need a new design of the entire engine?
Short answer: No, lots of them don't use oxygen at any temperature or density
Long answer: Less no. If you take an engine that uses liquid oxygen and put subcooled liquid oxygen into it, it will be subject to additional thermal stresses and the turbopumps involved will be pumping a liquid denser than they were designed for. If neither of those problems destroy it or prevent it from starting, it will work but the turbopumps will almost certainly not be delivering the oxygen at the correct rate because of the increased density.