From the YouTube app (on your DroidBOX device), there should be a Cast icon when a video is playing (if it detects any compatible devices on your network at least, not tried without any present). Click that and then the chromecast or other compatible hardware should be listed.
If nothing there:
1) Update YouTube.
https://droidbox.co.uk/how-to/first-run-of-googles-play-store/
2) May as well try updating the firmware on your devices. Open up the list of installed apps (circle with six dots, or "My Apps" label on an icon), and find OTA Updater or OTA Updates. Follow the onscreen instructions. You may need to clear some space for the firmware file to download, and if using Ethernet instead of WiFi, look for an option to click top right corner, if the download won't start.
You may also find it is worth installing the Chrome browser on your DroidBOX from the Play Store.
Beyond that, it is a case of checking each application on your DroidBOX to see if it supports casting to another device.