Why does the Collector-Emitter voltage need to be ≥ 0.3 V?
If you are designing a transistor circuit to switch and to have low losses you would like Vce to be as low as possible. You would like the transistor to be well into saturation. For a relatively low current single silicon transistor something like 300mV is a reasonable measure of it being well into saturation. 

For example, consider the 2N4401, a common NPN TO-92 transistor. Here is the typical behavior in saturation: 



As you can see Vce(sat) of 300mV covers the useful range of the transistor. 

You could equally well pick 100mV for collector currents of less than 100mA, it's just a reasonable choice based on how real transistors behave. 

Neither may be realistic for power transistors. Here is the similar graph for the 15A 2N3055: 



As you can see, even at 10A and a forced beta of 10, you'd be lucky to get under a few volts Vce.