Suppose an ATST is pushing a Rebel figure, iirc the Rebel player gets to choose where that Rebel figure can go ("Any figures in its spaces are pushed into the closest empty space of its controller’s choice")

Let's say this is the map:

E E A A A B

E E A A A E

E E

E E

E=Empty space, A=ATST, B=Rebel

If I move the ATST 1 space to the east, are spots O, P, Q, R all fair game to place the Rebel figure on?

E E O A A A -->B??

E E P A A A

Q R

E E

I think only space "R" is eligible. Diagonally it is 2 spaces away from the figure's starting space, while the others are all 3 spaces away.

Other than that, yes, you are interpreting the rule correctly.

So basically if every square have the same distance away, the Rebel player may place it on any of them. Otherwise it's the closest square?

For example: E=Empty, B=Rebel, A=ATST

Choose the closest space. If there are more than one with the same distance, the controller of the pushed figure decides which of them. (O or P in your example.)