Since, you are specifying that the print out is coming at screen resolution, you must be having a non-postscript printer or a non-postscript printer driver. This is a major reason for getting High or Low resolution.
First of all, if it is not necessary to print from Quark then it is good becoz Illustrator has a software RIP built into the application which can print at HIGH RESOLUTION even to a non-postscript printer.
But if you have to take it through QuarkXPress, then try to get the images in TIFF becoz it will give good quality in the worst case also ie through non-postscript printers.
Note: QuarkXPress relies on printer driver and printer to process graphics. Since non-ps devices cannot understand the information in a postscript file, the output is a low resolution.
Illustrator processes graphics before they are printed, so it does good even in case of non-postscript printers
Also make sure that
while printing from Quark in the Print dialog box in the options tab, the picture option is not set to LOW RESOLUTION. change it to NORMAL.