save bool in nsuserdefaults

Box your BOOL value to NSNumber object and add it to NSUserDefault:

NSUserDefaults *boolUserDefaults = [NSUserDefaults standardUserDefaults];
[boolUserDefaults setObject:[NSNumber numberWithBool:isquiet] 
                     forKey:@"stringKey"];

Later you'll be able to retrieve that value as plain BOOL using -boolForKey: function in NSUserDefaults

No need to wrap it in an NSNumber, there are some convenience methods for this:

To set a BOOL, use:

[userDefaults setBool:YESorNO forKey:@"yourKey"];

To access it, use:

[userDefaults boolForKey:@"yourKey"];

[EDIT TO ANSWER YOUR ADDITIONAL QUESTION]

Not sure why you are using NSUserDefaults - it seems unnecessary for what you are trying to achieve? Here's what I would do for a button that can start/stop music:

-(IBAction)check 
{
    if (isQuiet)
    {
        // Play music
        // Change the button to indicate it is playing...
    } else 
    {
        // Stop music
        // Change the button to indicate it has stopped...
    }
    // Set your isQuiet to be the opposite of what it was when the button was clicked
    isQuiet = !isQuiet;
}