Don't use EPS as it is NOT supporting Color Management nor transparency.
Never open any PDF (nor EPS) in Illustrator which was not created in Illustrator.
If the printer wants vectors, only PDF/X-4 will leave them as vectors in any case. If you use EPS or PDF/X-1a or the bad bad PDF/X-3, transparency will be flattened and vectors could be converted into raster images.
If a printer does not accept PDFs, select a different one. There are a lot of good printers out there.