are there matrix web messengers that require non free software?

No one can prove a negative, but this page has an "All known clients" feature where you can select a licence:

https://matrix.org/ecosystem/clients

Probably the JavaScript files are free software, but not compliant with GNU LibreJS.

You can use nheko in Trisquel.

sudo apt install nheko

I recommend the instance unredacted.org (do not require email).

Even if one web client is said to be entirely free/libre software to the client (you), the nature of these web apps allows them to be changed without your knowledge and without the ability to inspect the differences before execution.

The above is true even for web clients provided by websites you trust depending on how the web service is deployed.

https://www.gnu.org/philosophy/wwworst-app-store.html

This is why I prefer clients that are desktop based and depends little on web-based graphics toolkit, but without sacrificing federation. Thus I prefer standardized and federated protocols like XMPP or perhaps a future migration to GNU Jami, which is distributed, but the "future" part was mentioned since GNU Jami as shipped in Trisquel's default Aramo repos still presents some issues and GNU Jami also doesn't work well on Replicant.