What is wrong with my recessed light wiring?
If you have a voltage reading between hot and ground but not hot and neutral, it means you have an open neutral somewhere between the bulb and the panel.  Since you were playing around in the fixture, that'd be the first place I'd check.  Remove all of the wirenuts and check the quality of the connections, both from the line as well as the socket.  If the fixture has a transformer there will be another set of connections to check (primary and secondary sides).