Here is a video comparing a Hornady die with its conventional expander to a Lee Collet Die that sizes only the neck (you can size the body separately with a Redding Body die). You can see the runout from withdrawing the neck over the expander. Squaring a die, assuming your press doesn't produce self-alignment, won't help with this because it is how square the underside of the shell holder deck keeps the case aligned with the expander as it pulls down on the rim that matters. If you get a case with a dented or uneven rim that doesn't bear perfectly evenly downward, you are out of luck. The only press I am aware of that attempts to correct this is the Forster Co-ax press whose sliding shell holder plates can slip to self-align. Its floats dies in a slot, retained by their lockrings, so they are free to self-align, too.
I've tried the o-ring trick on a conventional press without much luck. The problem is the geometry of die threads isn't really right for this. You would want the die threads to be through the middle of a ball joint that can realign the die mouth on the shell deck on the fly. Instead, they are triangular profiles that don't mate evenly when the die tilts, so they try to line back up with one another as much as with the seating of the die on the deck.
One thing you can do to help with pulling the neck off-axis to use a carbide expander ball. RCBS makes them, including one that fits Redding dies, too. The first time I tried this I got an immediate improvement. I think it is mainly because the contact area is shallow so the case can be off-axis with it without much effect.
Another thing you can do is size the mouth pushing down. If you think about the geometry of a shoulder, it will be stronger resisting something pushing into it (like seating a bullet) than it is resisting something being withdrawn from it. So you can get tools like the Lyman M-dies for rifle calibers and not only expand without bending the neck but additionally can add the little step at the top of those dies just a short distance into the mouth so bullets you place there for seating stay straight up and enter the seating die ram straight. This does a lot to mitigate finished cartridge runout.
