Question about Logitech Z623

> It would work with...

Yes, it would: these speakers have very basic analogue inputs. RCA/Cinch and 3mm jack on the woofer, 3mm jack on the little speaker that has the controls.
As long as the DAC has a standard analogue output, it works.

> This would increase the sound quality.

- It depends on the USB DAC itself (don't expect miracles from an ultra cheap thing on Amazon)
- the reference point you're comparing it to (a good quality DAC from Roland or Behringer Uphoria will probably have less noise at the output than a poorly insulated thing integrated into the motherboard of some Asian no-name laptop)
- what you expect (a professional USB studio audio interface at 1'000 CHF might be overkill for simple PC speakers, even if they're good quality logitechs)
- and the gadget you want to plug into it (there is no more jack on the latest Apple i-Machin. An external DAC on the lightning plug or via Bluetooth is the only solution to use it)