Question about we MY BOOK with Ethernet connection

Can I connect my book disk drive with an usb/Ethernet adaptor on my routeur and directly use it on my Internal network?

See the user manual for your My Book. I do not believe this is possible, but it depends upon the My Book feature set. Manual will explain the various ways to connect MB to network.

You can’t. You can connect the My Book to your router if your router has a USB port. Some router will have one or two USB port that you can connect an external hard drive to. Then in the Webpage setting your can set it for files share or streaming video/music.