If you conduct a search on Bing, while on the HTTPS URL, it will remain HTTPS and keep your searches secure. This is a new feature Bing quietly started supporting over the weekend.
Cyrus Shepard tested to see if the query data would pass from Bing SSL search to Google Analytics. It is known that the standard protocol on the web is to not pass query referrer data from HTTPS to non-HTTPS, but from HTTPs to HTTPs, it should pass.
I am not sure if Bing SSL will pass query data from HTTPS to HTTPS but it definitely should not pass from Bing SSL to a non-HTTPS site.
Microsoft has not announced any plans for Bing to go SSL yet but this may be a sign that it will come in the near future.