Just use a boolean flag to indicate if the sound is playing or not. So something like this might work:
private boolean isPlaying = true;
private void MuteButton_Click(object sender, EventArgs e)
{
if (StartUpMusic.IsLoadCompleted)
{
if (isPlaying)
StartUpMusic.Stop();
else
StartUpMusic.Play();
isPlaying = !isPlaying;
}
}