I was wondering if I can export my endnote references as a bib file that includes the abstracts. This does not seem to be the case for X9.3.3. But bib format should be able to handle abstracts.
You can edit your Output Style to display the abstracts.
First, select the Output Style that you normally use. Then, click Edit -> Output Styles -> [Your style]. From the left-hand column, find Bibliography, then click Layout.
Click on the Insert Field button on the End Each Reference With box. Click on End of Paragraph, then click on Abstract. This will add an extra paragraph line between the formatted reference and the abstract. (You may wish to play around with the spacing - I personally have two paragraphs between the end of the reference and the abstract.)
Click on File -> Save As and rename the Style File to create a new Style (e.g. Chicago 17th Abstracts).
Close out of the Style Editor, then find your new Style from the Output Style dropdown list (you may need to click on Select Another Style to open the Style finder.) You should now see the abstracts displayed under the formatted reference from the Preview Pane. And you should now be able to export your bibliography in this new Style, which will now include abstracts.