How to create square wave circuit
You can do it with a dual opamp. The left opamp is a relaxation oscillator - can adjust the frequency with R1 - and the second opamp just inverts.
The AC-coupling at the outputs creates the bipolar output swing you wanted. The total output swing will be a volt or two less than Vin.

Here is a simulation, with Vin = 10V:

LATER
@Reinderien correctly pointed out this solution doesn't swing from Vin to -Vin. For that we need a voltage boost.
The following circuit will do that. It again uses an opamp for the relaxation oscillator: this should be a rail-to-rail type to get maximum voltage swing. The square wave feeds a cheap voltage inverter generating a negative voltage nearly equal to -Vin. A BAT54S does the two diodes in one cheap package. Dual comparator LM393 uses this negative voltage as a reference, the pullup resistor determines the high voltage.

And here is the simulation (using available opamp and comparator in the library but just use an LM393 and any rail-to-rail opamp).