Switching Boost Converter only works while probing with o'scope?
This MOSFET driver is not intended to do synchronous rectification. So though it's possible, here it's not connected in the right schematic to do so. 

Explanation:
When the inductor current reaches zero, Q2 switch must be turned off so that current does not flow in the reverse direction. This would keep the efficiency high. What's missing here is zero-cross detection circuit to quickly turn off the Q2 just before the current start to flow backwards. See "Diode emulation mode" in MCP14628 datasheet for additional explanation. Note MCP14628 is for buck converter.
I'm trying also to figure out a cheap synchronous rectifier DC/DC design with MC34063. I'm using boost cap to get higher than Vout voltage for the zero-cross detector...

It's interesting what efficiency do you get with this schematic at different loads compared to the same but with Q2 replaced by Schottky diode ?