I tried to remove the strainer in our stall shower today to get in there and clean out the drain; the house is 50 years old (but new to us) and I don't think the strainer had been removed in a long time (if ever). It's about 3" in diameter, held in by two screws opposite one another. The screws turned, then the heads promptly snapped from the threaded portions. The strainer lifted up, revealing the two "headless" brass screws. When I tried to drill one of them out, whatever it was screwed into disintegrated (it's covered in sludge, so I can't even tell what material it is). I managed to get one new screw back in, but it's not much of a fix.

My question is, in order to fix this properly, will I need to have the tile around the drain ripped out and have a new drain put in, or is there some other way I could secure a strainer in place?

I tried to remove the strainer in our stall shower today to get in there and clean out the drain; the house is 50 years old (but new to us) and I don't think the strainer had been removed in a long time (if ever). It's about 3" in diameter, held in by two screws opposite one another. The screws turned, then the heads promptly snapped from the threaded portions. The strainer lifted up, revealing the two "headless" brass screws. When I tried to drill one of them out, whatever it was screwed into disintegrated (it's covered in sludge, so I can't even tell what material it is). I managed to get one new screw back in, but it's not much of a fix.

My question is, in order to fix this properly, will I need to have the tile around the drain ripped out and have a new drain put in, or is there some other way I could secure a strainer in place?

Yes there is. It's called a snap in strainer and can be seen at; http://www.ipscorp.com/wt_html/snapindrns.html
Drill the old strainer out and install a snap in strainer. To see more strainers paste "snap in shower strainers up on your search bar. Good luck, Tom

Thank you Tom! I will look for those. Knowing the *name* of what one is searching for is half the battle for a new homeowner. :)