Shooting here from my rough understaing....
I would think that a short fast (1-2 days) wouldn't require any juice as you won't be getting all the benefits (talking about fruit juice).
Generally (with some limitations) your body works with glucose as fuel, any other saccharide will be broken down into glucose so it could enter the krebs cycle to produce ATP (or stored as glycogen). (I'm not sure when fatty acids come in to play though).
So if you keep the blood sugar levels normal (or high) with juices you might not drive your body to reduce metabolism. It might reduce metabolism in the digestive system though (much less work than digesting a steak).
Another issue: for me it's hard to keep blood sugar levels stable with fruits/juices if I fast for 18 or so - Insulin hyperreaction I guess. I know that 30-120 minutes later I might start feeling hypoglycemic (but I might be an exception).
On long fasts I would think that after depleating all resources the body will start breaking down stuff that weren't meant to be used for fuel, so you might be left with compounds that might add stress to the body and make apnea not so healthy. Question is, how long is long?
Something which I am pretty certain about fasts is that your body's reaction can be trained (like in freediving). So don't expect your first fasts to workout the best, try putting some consistancy in them.
Adding another thing to the discussion:
I just recalled something about the krebs cycle, I think it might accept Glycerol as well if there is no other saccharide to break down and it works much slower. So in theory if that is the only energy source you have inside your cells you might be able to cut down metabolism quite a lot... but that would work only after depleating glycogen stores, and maybe being low on a few amino acids that have carbon groups on them as well... anyone thinks a glycerol diet might work?
Ah, and I have no idea where fat/fatty acids comes into play with energy production, my "model" for all the above asumptions is a unicellular organism.
Interesting topic!