Why is the Champion not recommended to be used by it self ?
The Champion can be used as a juicer or as a grinder by making a small change to the machines configuration, by using the stainless screen or the blank/solid plastic block.

The Champion by it self  is an OK juicer , but it operation  presents a few problems if used as a one step juicer.
1) Once the juice  gravity drains through the  stainless screen at the bottom port of the juicer it is difficult for the juicer to push  a semi dry pulp through  a cone shaped restrictive front nozzle.  The pulp is being circulated inside the hub until it is expelled. This creates friction, friction creates  heat. If the temperature of the juice  heats to about 110 degrees the enzymes start to die. Heat is the enemy of the juice, it kills the beneficial enzymes. You would notice that your juice is a bit warm and has chalky taste. But what you don't see is that it has less enzymes and nutrition's in it.
2)  The pulp that comes out of the machine still has lots of juice in it, you have no other way  of extracting, good juice is being wasted.

What will the hydraulic press machine do? Why is the two step juicing better?

If you purchased a hydraulic press machine, you must to convert your  Champion to a grinder;  simply by removing the stainless screen and installing the solid plastic blank that came with the machine.
This will block the bottom port and not allow the juice to separate from the pulp.
Now the  Champion only grind fruits and vegetables into a paste. Champion is an excellent  grinder,  because it tears the fruit /vegetable fibers apart allowing the fibers to release their juice.
Because the juice stays with the pulp during grinding, the pulp becomes  soft/  apple sauce consistency paste which  will easily flow through the front nozzle without being congested inside the grinders hub. The juice/pulp  will not heat up.
This paste will go into the hydraulic press machine  and the hydraulic press will do the job of separating/squeezing the juice out of the pulp.
The hydraulic press  does an excellent job by pressing the pulp/paste with a force of 4000 pounds.

Can the juice be stored?
The juice should be made fresh and consumed right after pressing.  If you allow the juice to sit around in air it will start to oxidize, water and heavier solids, sugar will separate.
If you want to take the juice with you to work, the best thing you can do is to put in a bottle and fill the bottle all the way to the top not leaving any air in the bottle.
Option 2 : vacuum pump top bottles are sold; it will remove air/create vacuum inside the bottle.
"But other companies claim their juicer will make juice that can be used days later."
Not true! It does not matter which machine or juicer makes the juice, $10 or $2000 juicer; all juice will oxidize if left out in the open , even in a frigerator. Can it be used days later sure, but it won't be the same juice as the one made fresh..