'Squash' (which implies squeezed juice, but never tastes like it) is the fancy name for 'cordial' (which is not 'cordial' as I understand is described in the sense of a cocktail mixer-flavouring. It's (traditionally, perhaps less so now) something chemically coloured and flavoured to be (in the case of orange cordial/squash) as orange in colour as it can be and as orange-like in flavour as it can be (as in, as much 'orange-like' flavour, rather than as like actual-orange flavour), not a syrup and usually close to water in texture (although quickly capable of making your hands sticky, should the undiluted cordial get on your hands).
(Regular blackcurrant cordial is very dark purple and as 'watery', but still needs watering down. There were particular brands of blackcurrant cordial, though, that were perceptively more viscous and tending towards being syrup-like, and they were the good stuff. Never having had freshly-squeezed blackcurrants, I can't say for sure what they were both attempting to taste like, but the latter type 'tasted more real'. Meanwhile lemon squash was 'chemical pale-yellow', by tradition, etc. And this was in the days before artificial additives ('E numbers', not that all E-numbers are artificial) were considered unwelcome, and also there was no "no added sugar" advertisement, because that wasn't the extremely attractive selling point that it is now... But ironically they now probably use 'artificial sweetners' more, to achieve that.)
'Punch' is a (casually, perhaps slapdash) mixed drink, possibly alcoholic. Possibly with real fruit pieces in it. (Whether or not cored-and-quartered apple slices sitting in a liquid that is a mix of fruit-juices ever infuses that liquid... perhaps the liquid infuses the apple slices? Either way, it seems like gilding the lilly...) For children, however, a punch would likely be a mix of whatever kid-friendly drinks there are, e.g. the contents of boxed orange juice (likely 'from concentrate') bulked out with whatever diluted cordials (including orange!) are also at hand. It might still have bits of fruit in it. Apples, probably. Oranges, maybe. Fancy versions will end up with pineapple. Pear was rare, but not unseen. The flavour of a kids' punch would not necessarily be better than a single-flavour drink, but would be novel enough (in the days before "apple and blackcurrant" became fashionable variations of cordial) that not too much of it would end up being poured away. (Besides, it follows the general rule of the buffet table - If there isn't too much, there isn't enough!)
'Just plain fruit juice' is (or pretends to be, probably having been concentrated at the point of origin and then de-concentrated again with local water for boxing 'as fresh') squeezed juice. It tasted different - and more genuine - than cordials (self-made-from-concentrate), etc. It would never have had bits of fruit pulp in it, to proclaim it as 'genuine', but be a perfectly homogenised (and probably pasteurised) filtered liquid, thicker than a normal cordial but thinner than the 'best' blackcurrant cordials, prior to their dilution-to-taste.
In the modern day, I rarely encounter the term 'squash' (I'd probably associate it more with the specific vegetable/fruit/whatever-it-is group that gets mentioned, like pumpkins, in imported US TV/film), and don't have any 'fruit concentrate cordial' in the house to check the current terminology on the labels. But these days it's all about the "with no added sugar". 'Punch', likewise, is a term I never really encounter (perhaps something I'd expect to see served in a party on some US programme involving teenagers, e.g. Buffy or The O.C.), not attending enough wedding receptions and other 'dos' to know if they feature much, these days, and in isolation I'd probably associate the basic term more with the basic concept of the fist-attack. I also don't have any boxed fruit juices (from, or explicitly not from, concentrate) in my house, right now, either, but I do buy them and they've become 'even realer', somehow. And often I get a choice of various blended fruits (kiwi and mango, orange and cranberry, etc). That's even before I get the opportunity to choose 'with pulp' versions. But the first 'with floating bits in' trend I recall was bottled 'pre-diluted-from-concentrate' still cordial-like drinks such as Orangina, where you were encouraged to shake the bottle before opening (or, you learnt to do if you forgot, swirl the open bottle around to set up a vortex) to lift up into suspension the shavings of 'real orange' that were the selling point. I don't know if Orangina still exists, though.