Fresh-squeezed juice tastes better because you don't have to do all the things that need to be done to make juice safe to ship long distances. The idea that this super-expensive way of getting juice "damages the nutrients less" is a bunch of ridiculous hokum, and I'm betting if you got fresh orange juice that wasn't cold-pressed you wouldn't notice a 3X difference between it and cold-pressed.
Unless you actually saw it being squeezed, I wouldn't consider it "freshly squeezed". This is also the first time I've seen anybody, outside of advertisement, describe Minute Maid as "premium" orange juice.
Over here some supermarkets have juicers pre-filled with oranges in their produce sections, you just put an empty bottle into it, push a button and you can watch live as it squeezes the oranges into your bottle.
Don't know if they exist in the US, but I specifically look for juice bottles that say "NOT FROM CONCENTRATE". I can usually find a few even in smaller grocery stores.
I think what was meant by "fresh-squeezed" is the not from concentrate kind.
I certainly wouldn't describe juice from concentrate as "fresh-squeezed", because it's been squeezed, boiled to reduce its water content, frozen, packed, shipped, defrosted, rehydrated and packed again before I get to drink it.
