A big problem with Google’s Custom Search engine is the code is not formatted for users on https websites:
You can see the two urls
http://www.google.com/coop/cse/brand?form=cse-search-box&lang=en are both insecure. This is even after providing Google with the secure version of the site where the ad-code would be displayed. When implementing the code and the results page, most users see a blank page or a search box missing the google custom search logo.
The simple fix for this issue is to change the http in the urls to https. This will stop any mixed content errors and users over your http and https site alike will have no trouble loading the searchbox or search results.
Hope this helps anyone! Ask any questions in the comments!