Google bundles Flash with Chorme (making it the only option for some things on GNU/Linux), however they have recently started phasing out Flash. As part of that, Chrome hides the presence of Flash to websites, but gives users an option to enable Flash on the site if the page attempts to use Flash. eHomeAffairs and SARS eFiling gives an error when Flash is not detected and doesn't attempt to load the content anyway, which means that the "Click to run" option does not work.
Recommeneded method: MyBroadband documented one method to get eFiling working.. For eHomeAffairs, the address to add to the list is "https://ehome.dha.gov.za".
Alternative, works on many more sites: Another option is to configure Chrome not to hide Flash from websites. This can be done by visiting "chrome://flags" in the address bar and setting the "Prefer HTML5 over Flash" setting to "Disabled" ("chrome://flags/#prefer-html-over-flash" will take you directly to the setting). You need to restart Chrome for the setting to take effect. The content will then load. (Tested on Chrome 60). On some sites, the Flash content may still be click-to-run, however Chrome seems to currently run it automatically on both eHomeAffairs and SARS eFiling when this flag is set. (Chrome will attempt to detect important Flash content and enable that automatically)
Update: The chrome://flags method stopped working in Chrome 61. Adding the site to content settings as being allowed to run Flash as per the MyBroadband article still works.