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My site currently does not show that box when I search for it using example.com
Do I have to have Google site search installed on my site for this box to show up in the search results?
See this discussion....
New Search Box Under Sitelinks
At the time the Search Box appeared, there was concern about whether it would help or hurt traffic and customers. Interesting now that it seems to be seen as a desirable thing to have.
Chances are, though, that there's still no consensus.