With highcharts, you have a built-in button to download the current chart (example: http://www.highcharts.com/demo/, this button:
I haven't done that before, but I believe you want to play with the index.php
file located in the exporting-server
folder.
By default Highcharts provides (for free) a web service but you can modify that and create your own web service for exporting, or do whatever you want with the chart. Look at these instructions which can be found here Export module:
"If you want to set up this web service on your own server, the index.php file that handles the POST is supplied in the download package inside the /exporting-server directory.