I expect the output at HO and LO without anything connected to be a square wave
Don't.
The MOSFETS need to be connected to see an output square wave? ... Do I also need to connect the inductor and capacitor?
Of course!
Here's how a bootstrapped driver is used in a sync-buck converter:
simulate this circuit – Schematic created using CircuitLab
NOTE: I didn't show the external diode because UCC27201 has it internally.