In browsers that apply default styles using a default stylesheet, you should be able to override the default by identifying the appropriate style property and setting it to what you want. But it may be that some browsers either don't use a default style sheet, or don't allow you to override the settings.
Using a script-driven alternative is often a poor choice. Most do not correctly follow default browser behaviour, they just do what the developer thought was a good idea, breaking the UI for some users. e.g. when using keyboard navigation in Safari, the space bar is used to select options. Script-driven replacement select elements seem to be programmed to only accept the tab key for selecting options, so some users get a broken UI.
Why not just use the browser default? Why must you remove the highlighting? Users get used to their browser and how it works, messing with the UI is a developer thing that just makes life difficult for users. Incidentally, I'm not a fan if such highlighting (I hate the modern trend that everything the cursor touches must do something), but it's there so I leave it alone.