Can I shoot down a drone?

In short: not legally.
Attempting to interfere with, damage, or destroy a drone has the same penalties as trying to interfere with, damage, or destroy a human-crewed aircraft. It comes with multiple state and federal felony charges and can also be classified as a federal criminal violation (18 USC 32) and even a terroristic act.

You’re likely to be immediately arrested. Even joking about it is a felony. If found guilty of either, you could receive a multiple-decade prison sentence and permanent loss of your right to possess guns. You could also be liable for injuries or damages.

In detail:

• The same exact laws that protect all aircraft protect drones. So, anyone shooting at a drone would be treated the same as if they shot at a real human-crewed helicopter or airplane. The shooter could be arrested and charged with dozens of severe crimes. A conviction of any one of these could result in a multiple-decade prison term and permanent loss of your right to possess guns.
• Property owners own the land, not the air. There is ZERO private airspace in the US, and for the same reasons, property owners cannot prohibit human-crewed airplanes and helicopters; they can’t prohibit drones. Only the federal government can prohibit, restrict, or regulate this airspace. A landowner has no right to shoot at a drone and would be subject to arrest, prosecution, and civil liability.
• If the shooter causes a loss of control and the drone crashes into a person (or people), the shooter could be charged with assault with a deadly weapon or even more serious crimes. If anyone should die, the shooter could be charged with manslaughter or even murder.
• Anyone causing damage to a drone would also open themselves to severe civil liabilities and penalties. Drone operators and anyone with damages can actively pursue these and all legal costs through civil courts.