Is it healthy and/or acceptable for children 10-13 to do a 10 day juicing fast, or "reboot"? My family is interested in the micronutrient benefits.

Tween cleanse . I would be very concerned about this age group doing a 10 day cleanse. This is a time it significant growth and for tat lengthy of time i would be worried they would not get nough calories to sustain their growth at that time. I would discuss it with your pediatrician in detail before embarking in this direction.
Woud not recommend. If the adults in the family would like to "reboot", great. I would not subject a child to this. I believe the risks outweigh the benefits for a child.