It appears that the example charts showcased on Highchart's website are created using SVG technology.
I tried to display their home page using WebSupergoo's online demo for ABCpdf, which relies on IE, but unfortunately, the charts did not render successfully.
According to the documentation, ABCpdf only supports a partial implementation of SVG based on the SVG Tiny specification...
However, the most recent version of ABCpdf now includes support for converting web pages using Gecko's rendering engine. Since this engine is capable of supporting the majority of the SVG Full profile, it might be worth experimenting with. Details about the specific features that are still pending implementation can be found on Mozilla's site...
To switch to the Gecko engine, you will need to include a line of code like the following:
myDoc.HtmlOptions.Engine = EngineType.Gecko;
Each rendering engine has its own set of HtmlOptions. More information about the GeckoSubset can be found here...
If switching to the Gecko engine doesn't resolve the issue, it may be worth exploring other export formats supported by HighChart. Have you looked into whether they offer compatibility with the HTML5 canvas element?
Another possibility to consider is whether your charts are animated and if ABCpdf is only capturing the initial frame.