What is this black cylinder on the side of my electrical junction box?
That is an obsolete thermal detector and shut off if the can light gets overheated. It is probably why your can light is not coming on because it has ceased to operate. You can no longer get a replacement since they stopped manufacturing it sometime ago. If you find someone that still has one in stock you can replace it, but I ran out of them years ago so good luck. The only thing that can be done is to install a new can with a new built-in thermal detector. This could be a little complicated and may not be something a DIY could handle since you have to purchase the right part which is not a stock item and rewire the box to make it work properly. Sorry I can't be of much more help.

Good luck and stay safe