I don't remember what the thread size is. I do recall drilling out a few stubborn, broken off ones in both Ithaca (NID) and in a couple N/Spls. I retapped the drilled out screws with a common 6-32 tap and that's what I used for a screw size as a replacement.
The tip of the retaining screw is turned down slightly IIRC, just enough to remove the threads from the diameter.
The screw is only there to keep the FP in place and from falling back out of the frame. No need to turn the screw in so far that it jams against the clearance slot in the shank of the firing pin.
Some slack and movement of the FP around the tip of the screw can be noted when the 2 parts are in place. The FP can rotate quite a bit one side to the other with the screw in place.
Dry firing the Ithaca/Nitro Specials is an invitation to a broken firing pin
I don't see any reason at all that a hex head or allen type head screw couldn't be used as a replacement as long as it was modified to fit right. Make sure any screw w/a head on it has clearance for the hammer when down all the way. There isn't much room in there.
What makes the originals difficult to remove is that once installed, the factory assemblers placed a center punch on the frame over the threaded area of the screw and gave each a healthy wack to stake the threads in place.
Some are staked right on the top of the screw itself driving a punch stake mark betw the screw and the frame.
Using the small thin slot of the headless screw to try and overcome that staked in place commitment and years of dried lube as well often results in shearing the slot out or the head of the screw right off.
Quite easy to drill out, re-tap and make a replacement.
I put them in place with a bit of linseed oil on the threads. It gums up enough to hold them.
I doubt they'd ever back out anyway as nothing is twisting them. Never had one come back.
But easy enough to take back out if the need arises for someone in the 22nd Century.