How do I turn off the hardware monitor on UR44?

It seems that the device is somewhat crippled outside of cubase. Cubase is the only application that takes control of the hardware monitoring and in any other DAW, you are forced to listen to the DAW monitor in addition to the raw hardware monitoring signal through the headphone jack. DSPMixFX does not have an option to disable monitor, on OSX anyway. Am I missing something simple here?

Your post is not very specific so hard to judge what the problem might be. It may be possible that you have loopback switched on though.

Loopback is off. The device, by default, monitors anything that has been plugged into any input. This monitoring persists in Reaper, but turning on the cubase software will automatically override this monitor and allow me to control it through the DAW. If I enable/ disable monitoring through Reaper, I hear both the hardware monitor and the software monitor output.

Perhaps you need to delve a little deeper into the mysteries of Reaper to find the right setup. The UR28M (wich is the same basic architeture) here in my studio services both a PC running Cubase and a Mac running ProTools. There are no such problems as you describe in ProTools - it (meaning the DSPMixFX) works as it should.

How in the world do you disable monitoring through dspmixfx? I am trying to play a bass with my guitar using a pitch shifter, and it is impossible to do with the raw unmodified monitor signal playing through as well. Simply turning off the monitor function in reaper does not turn off the raw headphone monitor. It does in Cubase, however.

Have a look at your signal chain. I am assuming your pitch shifter is in reaper. I will assume you’ve got your guitar plugged into say HI-Z input 1 of the UR. If you check the signal flow, this signal is split to Mix 1 and 2 as well as to the DAW over USB. The phones 1 is hardwired to mix bus 1 (Phones 2 can be selected between the 2 mix busses). If you just want to hear the shifted (i.e. processed in reaper) output which is controlled by the DAW fader (2nd from right on the dspMixFx mixer, then you also need to turn the fader for Input 1 all the way down (or better yet mute it). This will ensure that only the processed signal will be heard and none of the live dry guitar.
Hope this helps.