I assumed you meant the Cuban missile crisis since that is the only tenuous connection that can be made between "USSR acquiring nuclear weapons" and "end of humanity". If you generally meant "any nation acquiring nuclear weapons" then the statement would kind of make sense, but there is no reason to fear just the USSR, and not, say, the US, China, India, Israel, Pakistan, or North Korea. Maybe some information made it into the popular press about the USSR, but it is unlikely that there is something especially dangerous about the USSR compared to all these other nations.

And in any case, the genie is out of the bottle. There is not gonna be a situation in which a small group of nuclear powers endlessly bomb and attack other countries with impunity. The other nations will get nukes to defend themselves against the existing nuclear powers, it's just a matter of time.