Clicking is quoted just for describing that image must be separate from background. Therefore it's assumed you don't actually need a click action. You merely want to ensure the images are separate from one another and any background.
If this assumption is incorrect, please clarify by editing the question.
Unless you add a background to the AI file, the exported PDF will have no background. PDFs, by default, show a white background because it's assumed that makes it easier to read the PDF in cases where backgrounds are transparent. It is similar to Illustrator artboards - by default they display as white even though they are actually transparent. There is no actual white there, unless you created it in Illustrator.
If you are indeed creating entire patterns in Illustrator and then exporting to PDF extracting the images can possibly be cumbersome. I'm not sure to be honest. In many cases, AI artwork is exported as vector objects, not "single" images. You may need to ensure you've grouped things correctly, or set up separate artboards (PDF pages) for each image in order for the C# to extract things properly. without knowing exactly how the C# reads the file, it's difficult to be definitive.
As for actual "clickable" images from Illustrator, you can't. Unless things have dramatically changed in the past year, Illustrator offers no interactive PDF features and does not provide a method of creating interactive elements within AI itself, and therefore can't export anything as interactive either.
If you need actual interactivity, then using InDesign, or Acrobat itself, to add that interactivity is probably the best option.