"Hello dear user, you probably know exactly what you're doing, but on the off-chance that you don't, please realize that disabling the POST option for https connections may leak your search terms to the visiting site".
Or something to that effect.
After all, then the referring url would just be the search page without any parameters.
So if you switch off the post then leakage would occur, with the 'post' enabled you're fine.
edit: I see what you mean now, if they switch to 'get' mode it leaks the info even to sites they don't visit. One more good reason to use that post!
For users who are annoyed, you could explain to them somewhere on the site that you don't put it in the URL because it exposes their query. If they really want a secure search, I imagine they'll understand the tradeoff.
They have a button on their website that adds it to you list of search engines in chrome (chromium in my case), so it definitely can be done.
EDIT: it turns out that ixquick switches to a different URL and uses get instead of post for this, so forget what I said.
PS: the opera urlbar inline search has post support.
Requests for:
http://duckduckgo.com/?q=hacker+news
Should HTTP redirect to something like:
http://duckduckgo.com/?enc=34g7h3giuh3g
Where 34g7h3giuh3g would be the ciphertext generated by encrypting "hacker news". That page knows what the search term was because it will have decrypted the parameters on the server side, but any referers would just contain "garbage", and it would also mean people can copy/paste the address bar about.
Referer: https://duckduckgo.com/