I have 2 screws that hold a shower drain that both rusted and broke off. Nothing at all to grip and they are small. Drilling it out will be really hard. Any ideas how to get the broken part out besides drilling? There is nothing to grab onto, nothing. If I mess this up my shower drain leaks into my finished basement.

Using a left hand drill bit is the only method I know.

Apply WD 40 and with some luck, it may back out.

I have over $5000 in drill bits and extractors, this is NOT the way to go. Instead purchase a new shower drain cover with 4 feet that drop into the hole/pipe drain. Spread those 4 feet to touch the side of the drain pipe and hold the drain in place.

Heat then allow cooling before trying anything else

Is this just the screen over the drain? If so why not apply a silicone bead to hold it in place?

Heat can work unless you have too much plastic around. A little break free spray might work best.

A bead of caulk is my backup plan. Replacing the drain is a huge project starting with cutting in the finished basement ceiling to access. Next cutting into hand laid tile shower floor. I was able to get 1 out. I very very carefully drilled a little on 2nd one - more to get it open to help speed up rust so it breaks free in the future.