as Alex says this thread needs to be in the discussion section.
I'm afraid the answer to your question is no. Freewheels are a screw-on (threaded) fit, where 10/11 speed cassettes slide fit onto splines and are secured with locknuts. There are lots of other issues too such as:
Rear stays will need a gap of 130mm for 10 speed, so the chances are that your frame will have to be altered by increasing the gap (can be done with steel but not aluminium)
Derailleur will need to be updated as a 6 speed one won't shift that far
Controls will need to be indexed; friction won't work past about 8 speed
You'll need a new hub; really a new rear wheel to accommodate the new cassette
Chain as you say
It's an expensive upgrade and possibly not financially worthwhile. What kind of bike is it? And why are you keen to move from 6 to 11 speed?