Do all A bodies have overheating problems?

Tune can cause over heating. What is your 340 build, timing total and advance and when. Carb and it a set up etc.

Bingo.

You said you have had numerous old cars so you probably know your way around one. I'm not trying to be a jerk at all but you could have been making the same mistake the last xx amount of years. This could explain why it's been happening for so long and why it never happened with the same engine in your friends car. In my profession I see old timers that are doing something completely backwards but they have been doing it for so long that it is routine for them. What is your tune and engine details?