Fresh juice has a lot of enzymes in it as well as a lot of various vitamins/minerals/"micronutrients." Especially if people's diets have been deficient in these things in the past, or if their digestive system is in bad enough shape that they are not absorbing nutrition from their food or from nutritional supplements, this can be helpful.
But there are a lot of other reasons that people cannot detox other than lack of nutrients or enzymes. So if there are other problems that haven't been addressed, the juice may not make much of a difference. Juicing also can be hard on the system, due to the various "anti-nutrients" in some produce (these generally become less problematic when foods are cooked) and the high amounts of sugar.
Some produce is quite detoxifying but only can be helpful if used in the right way, since it isn't enough by itself to get the toxins out of the system unless everything else is in place. One of the most temporarily disastrous things I've done for my health was to try a cilantro juice enema (and this was only a year ago, long after I got to "mostly well" status). I went from fully functional to basically bedridden within a couple of hours, and did not recover at all for quite a while. What I concluded was that the cilantro is very good at grabbing mercury and other metals from the tissue and bone stores, but that it does not hang onto the metals tightly enough to escort them all the way out of the body. FInally I started doing EDTA suppositories and the problem resolved almost immediately. How long it would have taken my system to dump all that loosened-up mercury without the EDTA, I don't know. But I definitely can see how even drinking cilantro juice could be catastrophic, for many people with this illness who have not resolved a lot of problems already.
When I had active CFS, I occasionally made juice and found that it felt like a good thing. I would imagine that nutritional deficiencies are problematic enough in this disease (due to digestive disturbances as well as challenges in preparing a healthy diet) that some super-concentrated produce might be really helpful. Substituting homemade vegetable broth (e.g. onions, celery, carrots, other stuff) for water was something that I felt like was equally good or maybe better, back when I was really sick.
More recently, as I've moved toward recovery, I've done phases with the juice. A lot of what I do is really intensive detox, interspersed with "building" phases where my body seems to crave as much nutrition as possible. Juice seems to be part of the building phase. I tend to do more juicing in winter (because juiceable produce such as carrots, apples, celery, etc. is more available then) and more green smoothies in summer (when I can get things like peaches and strawberries). Sometimes I go through stages of just eating cooked food. In general, I think that the body will tell you what it needs and that it's best to just listen to it.
While I've yet to meet anyone who said they got better from even mild CFS through juicing, I've encountered enough people who credit partial or even full recoveries to Gerson that I think this may be worth looking into. Gerson includes 13 glasses of mild juices (carrot, granny smith apple, mild greens, orange); 5 coffee enemas; a goodly amount of cooked vegetables; and a few supplements. I am suspicious about this because it doesn't include any protein (originally raw liver was included, now just liver shots) -- CFS sufferers seem to do better on meat, perhaps because trichothecenes act as protein synthesis inhibitors. And it really is an enormous amount of work (and a fair amount of money) to do Gerson -- having a well person to do all the food preparation is pretty much mandatory. I've never made the effort to go full-tilt on it myself. But still, it is really intriguing. So If anyone has any reports o CFS patients who have tried it, please send them my way.
Perhaps I should add here that I wholly agree with Gerson that all produce to be juiced needs to be organic, since concentrated chemicals are the last thing we need. The "Fat Sick and Nearly Dead" people only suggest organic but don't say it's mandatory, but they are mostly targeting people whose problems stem from crappy diets and whose detox mechanisms are basically intact. Gerson deals with much sicker patients (such as cancer) and thus is much more focused on the downsides of what modern chemicals can do.