Highchart hide default buttons

you can access each button preference with something like this:

exporting: {
    buttons: {
        printButton: {
            symbol: 'circle'
        },
        exportButton: {
            enabled: false
        }    
    }
}

an expandable example with your custom button would be:

exporting: {
    buttons: {
        printButton: {
            enabled: false
        },
        exportButton: {
            enabled: false
        },
        custom: {
            symbol: 'diamond',
            x: -62,
            symbolFill: '#B5C9DF',
            hoverSymbolFill: '#779ABF',
            _titleKey: 'printButtonTitle',
            onclick: function () {
                alert('click!')
            }
        }
    }
}

For anyone else who is using a newer version of highcharts and the selected answer doesn't work, you need to use the below instead to hide the button.

exporting: {
        buttons: {
            contextButton: {
                enabled: false
            }    
        }
    }