A few months ago I had an issue with LaserJet 1020 printer which was installed to XP and Windows 7. It was installed to XP and working, but when I tried to connect from a Windows 7 PC, it would attempt connection and then fail. Sometimes the Spooler Service stops. Sometimes, it can not connect -- items show up in the print Queue, but would never be printed.
The official driver can be found at the HP site for all OSes but (at least in my case) I could not get it to work. Here is the HP driver page:
I found more solutions related to this, for instance:
...but they are only partial solutions and workarounds.
I have found an old XP driver on the CNET site:
And (for whatever reason) it works perfectly. Even on Windows 7 it also works without any issue. Everything works fine now. I hope this will work for you, too.