What power rating of resistors would you use in the application required it to handle
0.6W?

Go with 1/4 Watt resistors but be prepared to use 1/2 Watts if the leads don't press down cleanly into the breadboard which is what I think you are asking about. If you need anything in the higher wattage range and you are working with an Arduino, then you are doing it wrong. An Arduino could not smoke a 1/4 watt if it tried.

I have fought for years with our purchasing agent over this at the college I work for. He buys 1/8th watt resistor in bulk, but fails to heed the advice that you cannot use these on standard classroom breadboard without bending the leads at best and failing to even make a good connection at worst. You need reliable connections and 1/8th and smaller are useless on breadboards. Just my two cents from years and years of

that is probably false!