Sorry, but this is not true. Airspace above private property is considered public right of way, in the same way that a sidewalk in your front lawn is public right of way. The ceiling of what belongs to the landowner is a gray area of law, but is generally considered to be the space that is actively used. If you have a trampoline, for example, then your active usage is slightly higher than that of your neighbor without a trampoline.

The FAA operates under a white-list. Everything is illegal until a regulation is made to make something legal. Radio-controlled aircraft have an exemption for recreational flight. Unmanned aircraft operated for recreation fall under that same exemption. For commercial, or civil usage, a certificate of authority (COA) is required on a per-case basis since no blanket regulations exist yet.

"That poses the question whether we are within our property rights to shoot down any drone flying around inside it?"

No. If someone drives an R/C car along the sidewalk in your front yard, you are not entitled to shoot it because of your property rights. At best, you could try to prosecute for trespass. Same with a flying vehicle, but unless the flying vehicle is under 20ft or so, the odds are not in your favor.