I never had problems burning 27C160s with my old (v3) Willem... I've upgraded to a GQ-4X (only for the USB), but I haven't tried it with that yet. I assume you've erased the chips w/ a UV eraser?
If you do a read several times in a row, does it always return the same values? Usually, if you do a blank check with no chip in, it'll pass... try that. A blank check is simply a read, so it sounds like either there's a problem with your circuitry (Data lines) and/or the EPROM (the VPP shouldn't matter yet, and if a chip is blank, your address lines won't matter). Do any other types of EPROMs work?
If you burn a file of all 0s (disable the automatic blank check before burning), then re-read it, you should get all 0s. Of course you'll then have to re-erase the chip, but that'd tell you a lot.
DogP