I did some "home surgery" again myself.
Dug in and broke some shards? maybe bone.. with a tweezer.
Had a good grip, but the thing won't even budge, so I just broke some pieces out of it.
It's still there and visible, but it doesn't hurt, so I'm just waiting now.

 

You are going to get infected and die.

 

Thanks for the concern, but not worried
Prescribed antibiotics and antibiotic mouthwash really helps.

 

It's not like he was digging in his gums with a rusty nail. A ~clean pair of tweezers or dental hook, and some antiseptic afterwords is really NO different than a dentist pulling it out. Only real difference is that he has a much better point of view to do a cleaner job. Gum tissue is a LOT like skin tissue and the connective tissue inbetween. You could go oldschool like they'd do with open wounds, get some finely grained sea salt and try to apply it in the hole with say an anti-septic wipe (to avoid germs from your finger). Hold it for a while, and see if it will hold onto the tissue. But like I said that could possibly hurt like hell.