Search folders don't actually contain items. They are a representation (shortcuts) of the search results. So, there isn't anything to export, per se. Therefore, you can not export them.
I did find this work around in order to save search result items to a
PST file outside of Outlook, provided you have permission to create and use Discovery Mailboxes. I don't have access to Discovery Mailboxes, so I was not able to verify this. It's worth checking into if saving the search results is important to you.
Use Outlook to Open a Discovery Mailbox
Export Multi-Mailbox Search Results to an Outlook Data File (.pst)
As far as Exporting Rules (you should have asked in a separate question), an easy search lead to this being the first result - Import or export a set of rules.