I think the big thing is that offensive Pokemon with Berry Juice become imbalanced (slightly better, if you will) but not broken. I played a game a while back using Berry Juice on my hazards Dwebble, and it got what, one more layer of hazards up than normal? This was neutered by the fact that everyone *had* to run Staryu in order to get their Sturdy sweeper safely in and blah blah blah.
Heysup's point about 2HKOs becoming 3 or even 4HKOs applies, but it also applies to Eviolite. There are more than a few would-be 2HKOs (especially in the case of mons with super offensive stats, such as Ponyta) that become 3HKOs or 4HKOs when the opposition has Eviolite. Does that mean Eviolite is broken? No, it just means Eviolite lessens the foe's offensive presence. Berry Juice is basically a counterpart to Eviolite; instead of lessening the foe's offensive presence, it helps the user's HP presence. To put it simply, an attack hits for the same damage whether the target has Berry Juice or not. However, provided the attack does not OHKO, the target can take more of these attacks with Berry Juice. Eviolite's useful in that it can prevent certain OHKOs.
In other words, I feel that the Berry Juice argument only gets dicey when one considers set up sweepers. IE, Tirtouga with Berry Juice is not only scary, but demanding for the team (your team now NEEDS a spinner, which, in this meta, means Staryu) and demanding for the opposition (the opponent now needs hazards and a spinblocker in order to keep Tirtouga off the field for longer).
The question here is not if Berry Juice is broken, by itself, arguing Berry Juice is broken is like arguing Eviolite is broken because it makes Scraggy too much of a threat.
The question is what does Berry Juice break? In my personal experience, I think Heysup's numbers apparently argue that Berry Juice breaks offensive Pokemon, especially those with a set up move that have enough bulk to take an attack, and even more the Sturdy Smashers. In my experience, most offensive Pokemon apart from the Shell Smashers are not broken after one turn of set up. For example, Scraggy after one turn of set up is honestly not that scary; it would always still use Eviolite. Misdreavus after one turn of set up is only scary if it is running Life Orb, so Berry Juice is a null argument here. The Shell Smashers (really, the only ones I'm considering are Tirtouga, Dwebble, and Omanyte; Clamperl needs either DST or Eviolite based on the team) are really kinda odd. The first two, Tirtouga and Dwebble have special note due to sturdy. Basically, you have to be careful because if you take out an opponent's Pokemon with something Tirtouga or Dwebble sets up on, you now have to do some gymnastics to get around the opponent getting a guaranteed 2 Shell Smashes. Omanyte's worth a mention since it actually beats a lot of Tirtouga's typical counters (Ferroseed and Lileep come to mind, as both can be "handled" with Ice Beam) and is bulky enough to take a hit.
So, it comes down to what's more preferable. Is the more preferable meta one that has no Berry Juice, but nerfed Shell Smashers (the current meta)? Or is the preferable meta the one that has Berry Juice, but banned Shell Smashers? That's right, I think the initial discussion should not emphasize Berry Juice as a whole; Berry Juice is not a "broken" item in and of itself, it depends on what you do with it.