No, supercapacitors could not withstand lightning. Very few things can, actually. You're dealing with millions of volts here; supercapacitors are usually rated for less than three volts, and even the highest-rated capacitors on the general market are rated for a few tens of kilovolts.
Additionally, there really wouldn't be any point in this, for the same reason it's not practical to generate energy from the wind in a hurricane: you're relying on an inherently random event occurring in a specific location where you've set up your infrastructure. Lightning happens more often than hurricanes, sure, but consider the cost of setting up a lightning-attracting structure (expensive) and a massive bank of capacitors (very expensive), and the buck converter to get that into a usable form (also very expensive), only to get maybe a few dozen lightning strikes a year.
While trying to find information on how much energy a lightning strike contains, I found that wikipedia actually has an article specifically on this topic. It seems there has actually been some research in this direction with, surprisingly, some success in laboratory conditions, but not anything too promising. Stick with solar power.