I did a lot of shipping when I owned a store. UPS technically requests new boxes. The concern is that used boxes may not be sturdy enough. I never had any problems. Most boxes use a certain percentage of recycled materials. Recycled fibers are shorter and make the boxes less strong, but still conform to needs. Reusing the boxes is better than just recycling. Recycling takes energy and fresh material.
Your shipping costs are a factor of the number of boxes, the weight, and the girth (dimensions). If you tape two small boxes together, the cost is less expensive. Don't use string as this can catch in equipment. Include to/from information INSIDE the box in case the label is destroyed. And if the box looks shoddy, recycle it.
UPS has no restrictions about how a box may be branded (the label). Just consider your needs and the image you wish to have. Remember to blacken out or cross out any old bar codes....I've had more than one box that I've sent out for delivery come back to me when someone scans the bar code that had my address on them.
The Post Office may require that you use their boxes for certain services for the best price. The boxes fit in automated equipment and provide information about costs and the service. Most don't realize they can get these boxes for FREE. You pay for them at the Post Office. Go online and they are free and they deliver them, too! (Some items have small charges.)