It's somewhat strange.
http://h10025.www1.hp.com/ewfrf/wc/softwareCategory?cc=us&dlc=en&lc=en&product=1103491& is the driver download page. Drivers are available for Windows XP, Windows Vista 32 bit and 64 bit and Windows 7 64 bit. Not for Windows 7 32-bit (although it's a very common version of Windows 7). You don't tell the bits-version of your Windows 7.
Try the printer install wizard on http://h10025.www1.hp.com/ewfrf/wc/document?docname=c02691615&cc=us&dlc=en&lc=en&product=1103491&rule=7118
It might find a driver.
If not, you can see if Dell is able to supply a 64-bit version of Windows 7 for your PC. Or you can call HP again and ask if it's really correct they do support Windows 7 64 bit and don't support Windows 7 32 bit. Or you can buy another printer.
That's about it, I'm afraid.
I wonder if someone can please help me, I have an older HP PSC 1401 All In One (printer/scanner/copier) that works for Windows XP & Windows 2000 from when I had my older HP desktop. Then I got a new Dell (Inspiron 560, Windows 7 Home Premium) a couple of years ago and never hooked up my printer as I didn't need it right away. Time went by and I tried printing one day I got an error saying it wasn't installed; I remembered that I never installed it on this Dell PC when I set it up. I soon found out that it was 'not compatible/supported' and of course this was after (it figures) I had just bought new ink for the printer, so now it sits with brand new ink cartidges (already opened so can't return) and can't connect to my Dell. I looked all over for help then after over an hour with them HP finally told me there is no way that their printer is compatible with my Dell (I just don't buy that.). I can understand if the OS requirements aren't compatible. I can't afford to just buy a brand new printer right now so any help is greatly appreciated. I also have a follow up question about ink if it's not going to work at all. Thank you!