If I had a small Cessna and I wanted to land at JFK international airport, would I be allowed to?
Probably.
What happened to you after that might be less pleasing.
Not really.
The answer to the OP’s question is yes, excluding military airports and small airstrips on private property. I have personally landed a Cessna 172 at Boston Logan airport, and it would be just as legal to do so at JFK.
If someone’s flying their own airplane from one country to another, aren’t they required to land at one of the airports designated to handle international flights before flying to another airport in that country?
You need to comply with the usual air traffic control procedures, and no doubt there are landing fees (which - I dunno - might be higher than you wish to pay, because the airport provides a lot of facilities that you don’t wish to use).
But if you’re happy to do these things, then why not?
You do need clearance from ATC to enter the airspace of a towered airport (in the US anyway, but most countries’ systems are similar), and they may not be willing to give it to you if you’re not on an IFR flight plan and they’re busy. You also need to comply with any special procedures and Notices to Airmen (NOTAM’s) that apply. Other than that, yes, you can land at any public-use airport. For private and military ones, you need prior permission.
I personally know about a half dozen pilots who have landed small, two-seat Cessnas at O’Hare airport in Chicago, I see no reason why you couldn’t do the same at JFK. As noted, there may be substantial landing fees. Also, they’ll have to fit you into the traffic pattern with all the much bigger and faster airplanes, which will not endear you to ATC. Those I know who have done such things recommend finding the “slow” traffic period and attempting the feat then - the O’Hare guys, for example. found the 2-4 am time slot to be optimum for their purposes.
THere are some military airports off limits to civilians, and post-9/11 Reagan National is likewise (commercial carriers only). Three airports around DC have heavy restrictions, but since those can be surmounted by someone willing to jump through the hoops, basically the only civilian airport in the US off limits to your small Cessna is Reagan National.
Even the ones that are off limits will probably (I assume) let you land in an emergency, depending on whether or not there were other airports nearby they could send you to instead.
Had a teacher who told us, if you’re a private pilot, and you want to land at DFW, you can, but it won’t be a terribly pleasant experience, due mostly to the high amount of air traffic you’d have to deal with , and yes, you would need to get permission from ATC before entering their airspace or else yousa in big doodoo.
Haven’t flown in over 10 years, but…
In an emergency, you can land anywhere. The investigation will probably be harsh depending on how badly you disrupted the ATC flow, and to what extent your negligence contributed to the emergency. (Wasn’t there some guy who landed on an aircraft carrier of the California coast during an emergency?)
For normal landings - also, IIRC there are requirements to enter some airspace - ie. a certain typ of transponder, altitude encoding? Plus, you’ll add to future grief if you do not understand the landmark locations they frequently reference; better have boned up on the airport neighbourhood from the published sources.
Finally, IIRC, permission to land - landing slots - are pre-sold to the airlines. If you don’t have prior permission during a relatively busy time you better have a real serious emergency. They pay for those slots, so expect a serious bill after putting down.
Having said that, outside of the really big airports like JFK, I imagine some have plenty of slack to accomodate smaller traffic - how busy for example is Clevland, Memphis, Charleston to pick a few lesser-rank cities?
The other problem, as mentioned above, is speed (and aircraft wake). The bigger the preceding aircraft and the smaller you are, the more separation required between their flight path and yours. Plus, the jets IIRC approach around 250 - 150mph and a Cessna at about 90 - 70mph.
For general aviation, there are a number of lesser airports that have customs officials specifically for private planes.
I’ve learned this from gliding.
And except for military bases, they really have no means to stop you.
And military bases probably won’t use every resource they have to stop you unless they perceive you to be a threat (it probably takes a lot for a Cessna to convince an F-16 to shoot it down, and making an approach on a runway probably isn’t on that list.) That’s not to say you won’t be met by the Security Forces troops on the ground though.
That said, if it was an in-flight emergency, and there wasn’t another appropriate airfield nearby, probably it will just turn into a good PR event for the base, about how the base personnel applied their skills to aiding a civilian aviator in distress. 
Even the military bases are often dual use military/civilian. Charleston and Savannah are two that I can think of off the top of my head.
While flying into the very large airports can be quite expensive my understanding is they waive the fees for Angel Flights where pilots of small planes volunteer their time to fly patients and families to medical facilities.
I think the largest airport I’ve flown into was Midway in Chicago. I followed a 737 on final, don’t think it cost too much, and the only real issue was they had to move the Pittsburgh Pirates charter jet because it was blocking me in when I wanted to leave the next morning.
Can a normal plane even land on an aircraft carrier?!
A Cessna wouldn’t be able to take advantage of the arresting gear…
What is the landing and stall speeds of a Cessna?
The carrier can do 32 knots, Cessna at 70mph is only 40 knots faster. (And this doesn’t consider any head winds.) The Cessna will have the length of the angled deck (which looks like it might be 2/3’s of the ships length, so guess 600ft) to brake to a stop, assuming, of course, we’re talking about the Nimitz type carriers…
Carriers have deployable nets that would stop it, though.
There was an incident during the evacuation of Saigon in 1975 when an overloaded Cessna O-1 was landed on the USS Midway:
Most military airports will let you land if there’s no other options for you, but you might not be allowed off the runway without a very thorough inspection/interview, etc.
Just make sure you don’t cause another crash on the way in…it might be a while before you get your plane back! In 2006, a damaged Embraer Legacy business jet landed at a military airport in Brazil after colliding with a B737, leading to it’s crash and death of all onboard. The pilots were arrested at the time, due to the collision, and the military/government held onto the Embraer for a very long time… it finally reached it’s destination in the US in 2010!
Wouldn’t it be redundant to arrest the pilots if everyone onboard was killed? Did tgey arrest the corpses?
Not sure if you were just engaging in some gallows humour here, but: the B737 crashed, killing all aboard. The Embraer made an emergency landing at a military airbase. Gol Flight 1907.
It’s the first time I notice that we have both a MikeG and a MikeS on this board. Now I wonder if I’ve ever confused them before…