CCM, DCM and BCM - Inductor
What is the factor that determines whether the power supply (dc-dc
converter) will be in the CCM, BCM or DCM mode?

The dominant factor is load current. If the load current goes beyond a certain value the boost converter will operate in continuous conduction mode: -

On lighter loads the boost converter will operate in discontinuous conduction mode: -

And, at exactly one point, DCM and CCM look the same - this is called boundary conduction mode.

What is the required inductor value that I need to choose for this
application which requires constant current for the LEDs?

If you want to run in CCM (probably the likely choice) then if the input voltage is 12 volts, the output voltage is 30 volts, the output current is 300 mA and the operating frequency is 455 kHz then, the minimum inductance value is around 12 μH: -

Note that I set the output voltage to be 30.7 volts to accommodate the forward drop of a real diode. The dotted lines in the above graph show the BCM point. If the converter current waveform fell below that BCM line, you'd be in DCM.
However, that gives over 1.3 amps of input supply ripple current and you may choose to make the inductor a lot bigger such as 100 μH: -

Now, the ripple current is 0.847 amps minus 0.686 amps = 161 mA
I expect somewhere between the two will be the right answer but, you can use the interactive boost tool to try out different scenarios.
Your value range from 39 μH to 91 μH is going to be about right. My low end value of 12 μH might cause the boost circuit to drop into DCM mode on a lower load current and sometimes, that can cause stability problems.