Electronic – use this inductor in the boost converter circuit

The MLZ series of inductors is not designed for power conversion (buck, boost, etc.), but instead for power supply filtering. This means that a) the windings are not optimized for low skin losses, and b) the magnetic core is not optimized for low core loss. From a circuit design perspective, this means the boost regulator will be less efficient.

If efficiency is not what you're after -- if you simply care for functionality and compactness -- then I don't see any problem using MLZ (or other "ferrite choke") inductors, as long as your circuit doesn't overheat. In practice, this means sticking to low power levels. Just ensure the current rating covers your needs (with healthy margin), and evaluate the power supply thoroughly.

Maximum output current is primarily determined by the current ratings of the switching FETs. Increasing inductance will only slightly reduce rms switching current due to lower ripple. So you could use a 20uH inductor, but it wouldn't significantly increase the maximum output current.

A more important factor is the physical size of the inductor. A 20uH coil with the same physical size as a 4.7uH coil will probably have around 4 times higher resistance and 4 times lower saturation current. Higher resistance causes greater loss. Lower saturation current causes higher ripple (sharply increasing as peak current approaches saturation). Both result in lower maximum output current.

So if you replace a 4.7uH inductor with 20uH inductor of the same physical size and construction the result will probably be lower maximum output current.