I am presently buying a shimano ultegra 6600 crank for my bike. My question is do I need to use the ultegra 6600 bottom bracket or can I use the dura ace 7800 as well? If I can use the dura ace bottom bracket is it a better bottom bracket than the ultegra?

You can use either BB. I'm not even sure they're different, since the prices I see for them are always the same. In general, you can mix 6600 and 7800 components:

http://bike.shimano.com/publish/content/cycle/sac/us/en/technical_service/compatibility_charts/road_compatibility.-MainContent-0001-DownloadFile.tmp/Front%20drive%20road%20compatibility.pdf.pdf

6600 is better according to some people here. in the last gen, dura ace BB's had problems in durability due to the gaskets giving way. ultegra are sealed bearings