Do i have the nut advantage as i have JJ+ in my range but vilain has 66 and quad ?

No, having JJ+ would be having equity advantage. BB has nut advantage. Has more 4x in his range compared to you but beyond that you have the stronger hands in JJ+ yes.

After his raise flop his range is pretty polarized

I don't think this is the case. You cbet small, his raise is small. This does not signify BB is polarized. Against a small cbet size OOP is supposed to raise often and will do so rather merged. 6x can take this line for sure.

i don't want to see an overcard on the river so should i bet or check

Bet. Both for value from 6x and denying equity from overcards.
But no need to be so scared of overcards. You do have position. A bet is still the better option.

Wich sizing ?

Think about what you are trying to achieve. Making BB fold his air hands that have overcards. Get value from 6x or weaker pocket pairs. A smaller size would achieve a decent amount of overcard folding. And forces BB to call more 6x and pocket pairs so he isn't overfolding vs our small size. A bigger size allows BB to fold weaker hands without making a theoretical mistake and you would be isolating yourself vs stronger hands (88+, 4x).

I'd bet 1/4-1/3 pot. It makes our bluffs cheap and a lot of our hands now improved so I expect a lot of folds from 53s 87s random over cards with BDFD that c/r the flop.

and a lot of our hands now improved

Do you mean that your overpairs became full houses? They haven't improved outside of 2 pair -> full house. If they were ahead they are still ahead. If they were behind they are still behind.
None of our air hands have improved.

The only argument you can make is that BB's range became weaker as this turn gives him less 4x combo's possible. And BB also didn't improve. Which reduces his nut combo's and allows our equity advantage to become the leading factor. Which allows us to bet small on the turn.

Yeah exactly, over pairs are now FH's so drawing to a straight isn't appealing, so I'd expect villain to c/f a lot.

Generally What you want to do is to size small when facing x/r, x line. Just use that as a default. You also can check back quite often.
When people xr They tell a story that They have a very strong hand, or a bluff. So the best counter is to size small with your range.
By betting small you still fold our all TJs typ of hands that xr with backdoors. And you loose minimum if he is going for double xr