Best option from Quark is to Export>Layout as PDF or save as EPS
Vector graphics etc. created in Quark are fully editable this way.
If using the pdf option, you may need to set the Output>Print Colors to DeviceN if you are using spot colors.
I would also use the Text To Box option to convert text to outlines wherever possible. One concern is how this affects your DCS files.
You may need to re-link everything in the new Illustrator file.