Some sites host an always-updated version.
Look for the latest stable build (usually labeled as an .html file, such as EaglercraftX_1.8.8.html ). Right-click the link or file and select Save link as... to download it to your Chromebook's "Downloads" folder. eaglercraft upd download chromebook
Eaglercraft is a reverse-engineered project. It is not endorsed by Mojang or Microsoft. While the original creator (lax1dude) made it as a proof-of-concept, many "UPD" versions are forks on GitHub. Some sites host an always-updated version