Hillary Clinton did not say that she would shut down the NRA and ban handguns if she were elected President.

Snopes

CLAIM: Hillary Clinton said that she would shut down the NRA and ban handguns if she were elected President.

  FALSE

"ORIGINS: On 25 October 2015, a Tumblr page dedicated to fake Hillary Clinton quotes posted a meme of the Democratic presidential candidate that included a quote supposedly uttered by her during an interview with the Des Moines Register on 8 August 2015:

"The "Shocking Hillary Clinton Quotes ..." Tumblr page claims that all of their quotes are 100% sourced and even provides links to make it appear as if the quotes have been verified:

This Tumblr page is just another fake quote generator created in the same vein as Facebook group "Stop The World, The Teabaggers Want Off." 
"Claim:   Sarah Palin told Sean Hannity Alaska has "all sorts of Eskimos and other foreigners.' " 

Hat tip to Karen Joy Roth