Barring some discrepancies, it could be It Conquered the World (1956).
One of several remaining members of its race, an alien from Venus is guided to Earth by disgruntled scientist Tom Anderson, who tells it which humans it should attach mind control devices to. Among them is his old friend, fellow scientist Paul Nelson. Nelson, after killing a flying bat-thing which carries the device, finally persuades the paranoid Anderson that he's been wrong to ally himself with an alien bent on world domination. They hurriedly leave when they discover Tom's wife has picked up a rifle and gone to the alien's cave to try to kill it.
Screenshots are linked to in case the YouTube video below goes dead.
Things that do match:
- 50s/60s movie, black and white, produced in the US;
- There's a spaceship crashing near a small town at 13'20. Command center thinks it's their malfunctioning satellite, so they reckon it's empty. Actually, the alien is in there, but only Tom knows that;
- there are mind-control devices attached to some of the city's key characters (by bat-like critters sent by the alien). The humans do behave "out-of-character" afterwards. The disks match your "50 piece, querter inch thick" description (see this blurry guy at 27'09 and this other guy at 32'04). You can see the "implantation" process at 19'38 and 22'07 for instance.
Things that kinda match:
- I'm not sure if disk removal leads to death, however it certainly doesn't fit with the alien's plans, as Tom says at 46'10:
Tom: His control device is gone.
Claire: What does that mean?
Tom: It means he must die. He's too great a menace to live the week.
- The movie ends with guns blazing, first shot is drawn by Claire at 1'00'38, followed by multiple shots from the military (1'06'17 and onwards). However, that takes place in the cave the alien was hiding in (not a spaceship, then) and the shots don't echo that much.
Found with the Google query
movie "back * necks" site:imdb.com which returned the reviews for Zontar: The Thing from Venus (1966). Several reviews mention this movie being a remake of It Conquered the World.