Method or Counter to track dropped connections in SQL Server?
To give information on the performance counter side...

The only performance counters I know of that might help would be the ODBC counters for disconnects. SQL Server has a counter for killed connections but those are connections killed by SQL Server, not the client.

You would have to capture these at the client which you could use typeperf command to do this remotely. SimpleTalk has a good article about using this command.

The comment suggestion would be the more human way of doing things.