In terms of the cheese, it doesn't care if it is pressed from the bottom up rather than the top down. The design may be more complicated due to the need to keep the platform aligned straight as it moves up.
But when you talk about a "new jack," are you talking about using a hydraulic bottle jack? I would strongly caution against this, for several reasons: 1) I'd be concerned about hydraulic fluids getting onto my hands and then to the cheese. (No such thing as a jack that doesn't leak at least a little bit, at least eventually!) 2) How will you measure the amount of force you are applying? (Certainly do-able, with a pressure gauge and appropriate math, but more complications and more opportunities for leaks ... and the typical bottle jack will not have a readily available port to add a gauge.) 3) Hydraulic jacks are really intended to apply 1000's of lbs of force, not the 5-10 you want to start with; it may be quite hard to keep the pressure down even to apply just the 200-300 lbs you'd want at the end of pressing a typical cheddar. (More pressure is not better - if you press too hard, too quickly, you will close up the rind before all the whey has escaped, and even at the end, if you press too hard, you will either get cheese squishing up around the follower and/or split your mold.) 4) If you just use a typical bottle jack, the pressure you apply to the cheese will dissipate as soon as the cheese compresses. So you will need, not only a pressure gauge, but also a pressure regulator and a pump that can continue to pump hydraulic fluid to maintain the pressure as the cheese compresses. Again, more opportunities for leaks and mess, not to mention you're now starting to talk about a good bit of expense.
Rather than a hydraulic jack, you will be MUCH better off using a pneumatic cylinder. Leaks are not a problem (so long as you are using clean air), and pressure regulators are inexpensive and readily available - though the less expensive, the less precise it will be. You will need to do some math to figure out how much force is applied to the ram for a given psi of air pressure - basically multiply the air psi by πr², where r is the radius of the pneumatic cylinder. But if you go with a pneumatic cylinder, there is no good reason to put it at the bottom; much easier to mount it at the top and press down. The only complication in going this route is having an air compressor at hand to supply the pressure; small compressors are readily available, run on 110v, and can be had for not too much money (under $100 if you go with a cheapie from Harbor Freight) ... but they are NOISY when they run. Really, really noisy. So if you're setting this up inside, this may be an issue.