This is the same sort of tubing you would see for a water line for a refrigerator ice maker.
It is 1/4" outside diameter (OD).
In fact, you can buy it at Home Depot:
Home Depot Tubing link
Removing and replacing the tubing on the diverter valve.
1. Remove the knurled nut.
Turn the knurled nut counter clockwise, looking at it from the tubing end, until it is free from the threads.
2. Pull the tubing off the stud.
This won't be easy. The tubing is being secured to the stud by a barb that is gripping the tubing from the inside.
You may be able to just wiggle it off, or you may need to slice it along its length.
3. Replace the tubing and the nut.
First, slide the nut onto the tubing with the open end toward the diverter valve.
Putting the tubing onto the stud of the diverter valve is a tough step too. Try wiggling the tubing as you push it.
Thread the nut back onto the stud of the diverter valve.
Removing and replacing the tubing at the AquaPurr.
1. Hold the collar, shown in the image, inward against the fitting.
2. Pull the tubing out.