I don't think the Square Wave Output is saturating the transistor
On the contrary :-) Through the 5k the transistor is always open. It might have difficulties to close, however... But it looks like an OC output..
I'll check in the datasheet
. Yes, it is.
You generally do not implement a (N)AND gate at the driver end, but at the control end, i.e at the base of the (driving) transistor.
In your scheme you will loose 2x0.4Volts at the two emitter collector diodes!
In this special case (assumed OC SQW), you can connect the buzzer-out directly to the gate of the first transistor T1. Also change the base transistor to the Arduino pin then to 4.7k (rather than 1k). T2 is no longer needed.
As you use the transistor for current gain only, you can also consider to put the buzzer between emmitter and ground (voltage follower circuit). Note that the circuit then is no longer in an inverted mode...