I want to build an interactive PowerPoint presentation in which I use buttons/hyperlinks to let the user move to the appropriate "next" slide. I want to disable the user's ability to go to the next slide using enter, page down, etc.

Is there a way to do that?

You need to enable KIOSK mode. In Kiosk mode, the only way to navigate is using onscreen elements. The ESC key is the only keyboard key that will still work, and it will exit the presentation.
Click on SLIDE SHOW > SET UP SHOW
Select Browsed at a kiosk (full screen)
Click on OK

If I set it up as Kiosk, will the user still be able to move from slide to slide using my buttons with hyperlinks attached? I thought that in the Kiosk setting, the slides just continually ran.

No, Kiosk mode only means "no keyboard". The presentation runs as designed. The only time a presentation runs from slide to slide on its own is if you set it up that way.