Is it damaging to drain an Eaton HS Supercapacitor below its minimum working voltage?

Eaton HS model is not a regular supercapacitor, but it is a hybrid supercapacitor.

It is a hybrid of regular supercapacitor and lithium-ion cell.

So same restrictions apply due to the lithium-ion cell technogy - the lithium-ion part of the hybrid capacitor will get damaged when it is discharged below safe voltage.

Boost converters have a high surge start current unless you design it to soft start with PWM or choose one with soft start built in. You will extend the life significantly of the Lithium oxide by staying within 3.3 to 3.7 V

Start with design specs for Zout and Z surge load for load regulation effects.