What is customs like for private planes? Is there some sort of physical control over the plane, its passengers and luggage until clearing customs?
As mentioned - if you fly into the USA, you need to meet the customs people on landing. If you yatch into the USA, you need to meet customs at the port. You cannot just fly in or out of any old private airport. (Maybe INS has a way you can pay for an immigration official to meet you at the airport of your choice, but I haven’t heard of it.) The obligation is on you to find the right place.
Sneaking someone into the USA is as bad or worse than sneaking into the USA yourself as a foreigner. At least a pesky foreigner would likely be deported. Americans have a way of teaching their own citizens to obey the laws.
Short version:
To legally fly into the US from another country, you
A) Must file a flight plan
B) Must file a complete crew & passenger manifest
C) Must land first at an airport with Customs ICE at a time they are open for business.
After you do those 3 things, and after ICE has had their way with you, your passports, and your goods, *then *you’re free to continue to wherever you want in the US.
There is a whole ginormous department of the Feds dedicated to ensuring everybody complies with all 3 steps. They have radars, guns, jet fighters, black helicopters (no really), and lots of ground troops.
Is it physically possible to sneak in via plane? Sure. Is it legal? No. Would an ordinary helicopter or airplane flown in an ordinary way buts sans flight plan be noticed & pursued & the folks arrested upon landing? Almost certainly.
Are there “special rules” for VIPs, which include the uber-rich? Not formally. But the customs folks at the airports where uber-rich enter the US have a different idea of customer service than the ones at JFK who bark orders at you.
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Also note that anyone with a private helicopter, jet and landing facilities also has a personal assistant (or a dozen) who would have the necessary passports, visas. manifest of good brought in, receipts for foreign purchases, itinerary and any other declarations all filed in a neat folder to hand to the customs people. As far as Bill and Carlos are concerned, they just take their sunglasses off and smile so the customs guy can compare them to their passport pic, and away they go.
Unless Carlos is persona non grata in the USA due to some of his business dabblings involving white powder and leafy green substances, he probably has already arranges one of those 10-year visas so any impulse visit to the USA is already cleared. That’s what he pays his assistants for… to make these arrangements.
This was the missing piece of information, plus what LSLGuy outlined below. I knew that there were some regulations regarding private flights, but I thought they were significantly different from those regarding commercial flight. Ignorance fought 
I wasn’t sure that a Learjet model could do France to NY … and when I checked it out on Wiki, it did appear that such a flight was out of range (a possible exception being the new Learjet 85s). There is a class of mid-sized “business class” jets, such as the Bombadier 5000, that can cross the Atlantic with fuel to spare and only require about 2700" to land. I’m betting that’s still very long for a private strip, but it’s better than 4000".
There are hiking trails between the US and Canada that are unmanned boarder crossings, one such is the international Appalachian trail (an extension of the AT). The general rule is you are suppose to go to the customs of the entered country as soon as practical.
I also did enter Canada via jet ski once which that rule also applied.
No, Bill Gates cannot fly into the country in his private helicopter without going through Customs. His ability to sneak someone in is the same as any other vehicle.
The Northwest Angle of Minnesota can nly be reached on land by going through Canada. So when you cross that border, you have to report in to Customs.
But there is nobody from Customs there – instead, they have a booth with a videophone connection to Customs which people use to report in. But there’s nothing there to stop you from just driving by the booth without reporting. (In fact, some local residents do so frequently. Their feeling is that if you report crossing the border once a day, that’s good enough.)
The visa waiver program was mentioned upthread. The visa waiver program does NOT apply if you are entering the USA in a private plane (or helicopter, or yacht…) Even citizens of visa waiver countries, visiting as short-stay tourists, require a visa unless they enter by commercial air transport or cruise ship.
Most people, unless they are Bill-Gates-rich, probably hire a jet or helicopter charter. A private jet for someone who only uses it a few times a year makes no sense.
So would you have to come in by regularly scheduled transport? After all, until recently a lot of tourist traffic was not commercially scheduled traffic but charter jets (although with a hundred or more passengers).
Seems like a lot of people are missing the point here: so apparently you need a Passengers Manifesto.
Maybe as an extension, if you enter to the states and then Slim jumps off the plane in a parachute, could the radar catch that?
I thought the Appalachian trail took you to Argentina, not Canada. 
The AT is from Maine to Georgia and a bit over 2000 miles long the international AT is a extension that runs into Canada, but is sort of unofficial as a continuous trail.
Some versions of the I-AT run from Key West Fl to the AT into Canada then over to Europe and Africa as the mountains used to be part of the same range, some European countries have recognized the I-AT though their lands. I have not heard any mention of including south america as well as the mountains sort of end in the US, but could see a international contential divide trail running through there.
I think this could work, but there are obstacles. Obviously, the jumper would have to be trained and experienced, but assuming he or she is:
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The type of aircraft is again a factor. Assuming a larger plane for an Atlantic crossing, it may be difficult or dangerous to open a door in flight.
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A larger aircraft is faster too, making a side egress dangerous (I’m assuming we don’t have a D.B. Cooper style rear exit). The aircraft could be slowed, but this might be seen as suspicious by ATC. Depending on the aircraft, it might not be possible to close the door again afterward, which while not necessarily dangerous, could cause suspicion if seen during landing.
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I know that ATC radar can sometimes plot things as small as birds, but I don’t know if a jumper would be seen exiting an aircraft, and if so, if he would be interpreted as such. Once the parachute opened that might present a larger radar cross section, but again it might not be interpreted as anything more menacing than a bird.
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I think the two real problems for the jumper are being seen visually by someone on the ground, and leaving the landing zone unobtrusively. This suggests a rural landing area, but that’s also more dangerous. While a general area could be guessed, it would be difficult to pre-select a particular spot with any assurance of the plane being routed over it. So the landing area would be a matter of luck and quick decision by the jumper.
This stuff is fun to think about, but it’s a good thing I’m one of the good guys who is not trying to get away with anything shady!
I’ll modify my point about routing the aircraft over a particular spot for the jumper. That could be possible, depending on the weather.
I don’t think an ADIZ flight plan for entering U.S. airspace has to necessarily be done IFR with associated routing. I think that can be done VFR, which gives the pilot much more choice of route. I’ll check on that, but even if they were IFR I think they could cancel and proceed visually once inside the airspace when conditions allow. The filed flight plan would still be in effect for the point of landing, but VFR instead of IFR. I’m not totally sure of this, but I’ve been in similar situations in restricted airspace where it went this way.
If that’s true, as long as you went on a day with decent weather, it might be possible to overfly a pre-selected landing zone for the jumper. Again, all this is in hopes of landing the aircraft legally at an airport with customs, with nobody the wiser about the jumper.
As a frequent flyer of commercial airlines, my opinion is that of course we need a Passengers Manifesto!
What that has to do with the current topic, I have no idea. 
Definitely unofficial, it’s not recognized by the Appalachian Trail Council. There’s no connection from the top of Baxter Peak to where the IAT starts outside of Baxter State Park so it’s not continuous. It covers some very nice terrain but it’s not really part of the AT at this point and likely never will be.
The AT goes all the way from Columbia, SC to Whoosh, Argentina.
Just ask Mark Sanford…
(Poor choice of smilie though, Gorsnak.)
The criteria for eligibility to travel under the visa waiver include: