I personally don't think it's healthy or realistic. The human body needs a variety of nutrients: fat, carbs, protein, etc. An all-juice diet is basically entirely carbs plus the juicer removes lots of the healthy, nutritious parts of the plant.
It's just not a realistic plan either. The human body is meant to chew solid food rather than ingest liquids exclusively.
Also my personal philosophy is that I won't do anything to lose weight that I can't see myself doing for the rest of my life. Permanent lifestyle changes lead to permanent weight loss. Temporary fixes, like a juice cleanse, will let you lose temporary weight.
You don't have to eat the whole thing.