> 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:

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![schematic](https://i.sstatic.net/VNmrZ.png)

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NOTE: I didn't show the external diode because UCC27201 has it internally.