From your description I asume that the button that pop’s out is the overheat circuit switch which shuts down both the pilot and the main burners in the event that the boiler has overheated or gone past it’s set temperature on the thermostat. The problem could be caused to lack of water within the system of the boiler or a simple air lock – best to have it checked out by a qualified boiler technician in your area.
Hope this helps.
Selgas Services Ltd
Craftsman Gasfitters, Plumbers, Electrical Service Technicians