Cougars in NY info:
Here's two links for cougars in NY state:
This link advises you on how to preserve the tracks & evidence of cougars
http://leahyinstitute.org/?page_id=436
This one has lots of photos:
http://www.dec.ny.gov/animals/44564.html
<mom_mode>Please, please don't get a shotgun, more because I would worry about your safety than anything else. You can get more chickens, but you cannot be replaced. These are very big predators and they wouldn't hesitate to seriously mess you up, shotgun or no. </mom_mode>
What you should look to get or borrow is a wildlife trail cam. Ask any hunters if they have one you can use. Even if you don't get shots of a mountain lion, you get to see who/what's in your yard at night which is really interesting. Get the reflective eyes you can put on your coop to fool predators.