Are push-connect sockets intended to be reusable?
It depends on the light socket, and maybe how you "disconnected" it.  Some push-in type wire connectors are actually threaded; they grip when you just push them in, but you can back the wires out by twisting them (lefty loosey).  However, if the wires were yanked out, it's possible that the threads were stripped and won't grip any more.  I assume you have also checked for small screws. Sometimes connections that look like push-ins are actually clamps.  You stick the wire in the hole, and then clamp it down (internally) by tightening a screw.