Why is the square wave signal so distorted at the output of a push-pull pair?
Think about what the load resistor is doing. Without that resistor, how are you going to put base current into either transistor and turn Q2 and Q3 on properly. With the resistor, Q2 can be effectively turned on and that same resistor acts as a decent pull-down when Q3 is supposedly being activated. Without proper base biasing you won't have a decent push pull stage. Try using a 10 kohm in parallel with collector/emitter on each transistor to see what happens. Or, alternatively try biasing the bases as per how a class AB stage operates.