I've scoured the message boards and tried a bunch of stuff here, suffered the remainder of the HP "support" site, and I'm about ready to throw this printer away and try a new brand. Thought I'd give this one whirl instead.
After working fine for years, HP DeskJet 3504 J610a keeps flashing popup messages warning me that the printer files (all kinds of names) "will damage my computer." Yes, I know - we've seen this before. So I've done the following:
* restarted the computer
* removed the printer from the System Preferences list (and re-added it)
* visited this site looking for an answer, tried the non-functional virtual assistant (clicking doesn't work) and the tease message describing my problem (clicking doesn't work)
* logged off my browser and tried the site again
* tried the non-functional virtual assistant (clicking doesn't work) and the tease message describing my problem (clicking doesn't work)...by the way clicking DOES work on other parts of the site; I do know how to click my mouse if that's in question
* found the message information another way, elsewhere on support site
* un-installed all HP software, using the procedure finally found on the support site
* looked for updated drivers like the procedure said, but found none listed for the J610a (most recent is Mountain Lion in 2014)
* searched the site some more for other browser libraries, found nothing
* scoured community boards and took advice to use a similar driver, used similar driver, failed
* un-installed all HP software again
* restarted the computer again
* connected wirelessly to the printer from scratch and had it download the updated driver software automatically
* un-installed again after having the pop-up problem happen worse than ever, and unceasingly now
* restarted the computer again
* did the last three steps all over again to be sure
* visited the community again and looked over all "similar messages" to make sure I wasn't missing anything
* logged off with the intention of trashing the printer and never buying HP again, and then on a lark decided to post this last attempt for help.
Holiday shopping for a new printer tomorrow, if no one else has any ideas. Thanks in advance, stay safe all.