This is static content. It has nothing to do with form fields. These boxes don't actually exist in the PDF. Acrobat generates them for you to facilitate editing the contents of the file, but you should really not be doing that if you have access to the original file format. If you want to make changes to it do so in AI and then create a new PDF.
If you've already added fields you can then use the Replace Pages command to replace the old pages with the new ones, which will keep your fields in tact.