Yes, you can run both the HERO6 Black and HERO7 Black with external power. Same with the HERO5 Black and HERO (2018). If you're powering the camera from an external source via the USB-C cable, whether that's a wall adapter, USB powerbrick, computer, or car adapter, you can remove the internal battery and run the camera via external power. The camera will continue to work normally.
An advantage of leaving the internal battery in place is as a safety net in case the external power stops flowing. Without the internal battery, the moment that the AC power stops, the camera will turn off. It won't turn back on automatically when the power source is restored. With the internal battery installed, it will switch over seamlessly. But there are times where removing the battery might be a better option, especially for very long-term shooting such as timelapse.
External Power Indicator
Usually, when you connect external power you'll get the charging battery indicator. Without any internal battery to charge, though, there's a different icon on the front screen. It looks like this:
On the HERO5 Black, HERO (2018), and HERO6 Black, you get the same icon in the top right-hand corner of the back screen as well. On the HERO7 Black, you get a small "No Batt" message next to a red battery icon with a cross in it.