Am I better off buying the $300 brand new Yamaha/Fender etc. straight from the shop (they've got a sale at the moment) or spending the same amount of money on a 2nd-hand instrument that was slightly better made?

I've only ever owned a beginner's electric. I'll be using the acoustic for learning/practicing and songwriting but probably not for much performance.

A related question - if I've been playing a Squier Strat for years and not minded it at all, am I likely to notice that a Fender acoustic is an inferior guitar?
For the price, I'd go with the Yamaha as they are generally pretty good guitars with superior construction and components (solid top, better hardware) for the cash.

If you can find, try out and fully inspect a used acoustic then it's a good buy. Look for solid woods as opposed to laminates. A used guitar with solid wood construction needs time to "bed in" as the wood still has natural shifts and the tone will mature - you wouldn't have to wait as long to get a more matured tone from a used instrument. If you try out a used instrument and have a smart phone google for reviews of that model and also online sales to see if the price is reasonable and to see what the general consensus is regarding playability and tone.

The usual things should be looked for - straight neck, clean joins, any repair work carried out. Just be sure that you're not rushed by staff in a store regardless of what you decide.
It's an opinion. It's subjective. And I'm right, anyway.
I'd go with used if you can find one you like. If you decide to go new, a Yamaha FG730 is pretty hard to beat in the $300 range. Others to consider are Seagull, Washburn, Alverez, and some Fenders aren't too bad.