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.