I have an 8 speed Specialized Secteur and plan to go to 10 speed this Spring. The hubs are Shimano Ultegra 6600. I found a Shimano compatibility chart but there was no listing for 6600 hubs. So the question I have is; will 10 speed shifters and cassettes work with this hub?

Yes.

Yes. Ultegra 6600 is 8/9/10 speed. See the 2009 docs at 2017-2018 SHIMANO Product Information Web

Shimano Ultegra 6600 was a 10 speed groupset, so naturally, 10 speed cassettes will fit a 6600 rear hub. In order to migrate to 10 speed, you will need 10 speed shifters that will work with your current derailleurs. The only current 10 speed group manufactured by Shimano is Tiagra 4700. It is pretty much certain that your current derailleurs will not work with them, so to move to 10 speed you will need new front and rear derailleurs unless you can source older 6600, 6700, 5700, or 5600 shifters

Last fall there were really good deals on 5600 components from the UK but that boat may have sailed. If not, Tiagra is a pretty great deal. I installed Tiagra levers, FD, and RD on my son's bike and I was really impressed. Not as polished as the higher groups, obviously, but works perfectly.

That is correct. Any previous generation 10-speed road shifters (brifters, barend or downtube) will work properly with a 10-speed cassette and almost any road 8,9 or 10-speed rear derailleur.

The exceptions are:
1)10-speed Shimano MTB components (shifters and rear derailleurs) are not compatible with 10-speed road shifters or derailleurs.
2) Current Tiagra 10-speed shifters and rear derailleurs are not compatible with older 10-speed road components.
3) Pre-9-speed Dura Ace rear derailleurs won't work properly with 10-speed road shifters.

You'll need to run a 10s chain too.

Thanks guys. Much appreciated.