One of my buttons for my rubber bands fell off, should I continue to wear the rubber band on the other side?

Hi, I would recommend waiting until the button gets re-attached to start re-wearing the elastics. The reason is that if you only wear the elastics on one side, the bite is only getting corrected on one side and can cause the bite to become uneven and not symmetric. Waiting a few days or a week until you can get the button put back on won't have a huge effect on your overall overbite correction with elastics.
So TL;DR, don't wear elastics until the button gets fixed and you can start wearing elastics on both sides again.

you should contact your provider ASAP, and get the button reattached. I do not recommend wearing it only on the 1side.