Removal is difficult without damaging the chairs. One possible solution is to dig around the screw with a knife or other scraping tool until you can get a grip on it with pliers or vice grips and unscrew it. Then use wood putty to close the hole; sand it smoth; paint it; and re-drill the hole for the new screw.

There is a basic tool called a "screw extractor". It is a hardened steel screw with a reverse thread. Your local hardware store will have it. If possible go a little larger when putting in the new screw; or put a small wood sliver into the old hold to give the new screw a good bite so it will stay firm.