Can I install a floodlight with 18AWG conductors on a 20A (12AWG) circuit?
Most likely.

As long as the fixture has a UL (or equivalent) stamp/sticker, there is no problem. 

Fixture wires, and the wiring inside devices is specifically designed to carry the rated loads. There are different rules applied in this situation, since the load is known and static.

According to the National Electrical Code 2014 402.5, the allowable ampacity of 18 AWG fixture wire is 6 amperes. Your fixture will only draw 2.5 amperes.

I = P/E = 300 Watts / 120 Volts = 2.5 Amperes

If the manufacturer could use smaller wires, they probably would. However, NEC says 18 AWG is the smallest allowed for fixture wires.

One of the reasons light socket adapters can be dangerous, is that they can allow more current to be drawn than was intended by the fixture manufacturer.