Hairs grow back white because of damage to the hair follicle that lies under the skin. Sometimes the follicles get damaged deliberately (for freeezemarking, where you WANT the hair to grow back white) and sometimes they get damaged accidentally through injury or disease. Dyeing the hair shaft itself will change the colour of the shaft but as new growth continues, it will still grow in white. This is why people who dye their hair keep having to have their roots done!
The only way to effect a permanent change of colour would be to do some sort of a follicle graft. You would need to capture the hair follicles that sit quite deeply in the skin and I guess the very act of that trauma might well make them grow back white anyway! Boot polish or one of the touch up sticks you can get for horses is your best bet.