When considering flying an aircraft across the U.S. border, you also have to consider that there is extensive radar coverage of all airspace within the U.S., with varying flight ceilings for different classes of aircraft. Monitored not just by the FAA, but also the Department of Homeland Security and the Department of Defense.
This isn't just a safety issue of resolving conflicting flight paths, maintaining air corridors, managing airport capacities, and avoiding collisions. This is also an issue of safeguarding national boundaries and preventing smuggling and terrorism by tracking authorized and unauthorized entries.
Entering the country by crossing into its airspace without filing a valid flight plan and having a valid visa/proof of citizenship would constitute a crime of illegal entry. Landing in an out of the way location without a valid customs zone could be construed as trying to avoid registering your presence to the U.S. government, declaring imported goods, paying taxes, or a more nefarious purpose.
The worst case scenario, depending on what the current DHS security level is, could result in you picking up an escort of F-16s to take you to a secured airfield where federal agents will take you into custody, do an extensive full body cavity search, grill you to the nth degree as to your origins and purpose, and detain you for prosecution on any number of charges that are based on what they dig up on you. And don't forget that the NSA has been collecting a vast amount of information on everybody.
Department of Homeland Security U.S. Customs and Border Protection: Pleasure boats and private flyers