Consider it no.
Can you wash birkenstocks in the washing machine. If your backpack has a metal frame remove this before washing. I agree with PerspicaciaTick about washing them in the machine. For hard to remove stains try to carefully scrub the affected area with a soft bristle brush but dont use any silicone-based cleaning products.
Cold water and a delicate cycle is the way to go when washing your shoes. Its crucial to remove all loose parts such as the laces and insoles. I put mine in a laundry net - a pillow-case would just be as effective so they dont bang around in the drum.
After that your sneakers are almost ready for the washing machine. The short answer is that it depends. Check the pockets for any small items and check the corners for any dirt that may be lodged.
Washing Birkenstocks in the machine could also damage the washing machine. Can you wash Birkenstocks in a washing machine. Can You Wash Slippers in the Washing Machine.
Empty your backpack and leave all pockets unzipped. Perhaps the most important step is. Now learn how to clean your Birkenstocks.
Air dry the shoes after. Obviously you want to try and remove as much moisture as possible so we would recommend having an extra cloth around to switch to. Even if you think its sturdy dont do it.