Under 'File' check that the document mode is set to CMYK. Instead of 'Export' go to 'Save a Copy' and choose 'Adobe PDF'. In the options dialogue box click on 'Standard' and select a suitable print standard such as PDF/X-1a:2001
I just did an 'Export' as JPG from Illustrator and checked it in Photoshop with the 'Info' tool and the colour shows as 0, 0, 0, 30. It looks grey on the screen, too.
I cannot replicate the problem so suggest you check that the colour mode is CMYK not RGB (under 'File' 'Document Color Mode') and also check you are using CMYK in the colour palette and/or swatches.