You are using UseShellExecute=false. In this context, the MSDN specifies that the FileName property should be a fully qualified path to the executable. So, just writing mysqldump.exe
as the filename is not enough to start the process.
I recommend to set also the WorkingDirectory to the path where the mysql binaries are installed because, if the process launched needs some local installed resources, withoud a WorkingDirectory set you could have hard times to understand errors about files not found or missing configuration files.