I've never found a way to completely clear a single box in a single operation, but if you highlight the entry you want to remove, you hit the "Del" key, and it goes away. This works in Firefox, not sure about other browsers.
I don’t use Google search much. In fact, I only use it as a last resort. Bing works fairly well, and I find that when I start to type something in, and the suggestion drop-down box appears, there is a “Manage Search History” link at the bottom right of the box. Clicking on that and clearing out things I don’t want prying eyes to see (like when I did a search at Victoria’s Secret’s site for Valentine’s Day - I didn’t want my wife to “accidentally” find out!!) seems to clear the drop-down box for all search engines.