Can a Non-EU citizen working in LU become a cross border worker?  

Yes you can do that, but the other national paper work must be done.

Hello thanks, any website or administrative office that I can find out more on the paperwork?

We need to know which County will be your next, Germany, France or Belgium? At best with postal code as it is most a local office which you need to contact.

Hello, thanks for answering. It will be Perl, Germany 66706

thanks so much!

Hello, we had a similar case with my wife, who is a non-EU citizen. We work in Luxembourg and live in Germany. Since I am an EU-citizen working in Lux, she did not need a work permit (see second link below).

The general work permit information is here:

Working in Luxembourg as a third-country national and cross-border worker — Citizens — Guichet.lu - Administrative Guide - Luxembourg (public.lu)

More relevant in our case was this very specific exemption:

Work permit exemption for third-country nationals who are the spouse, partner or child of an EU citizen cross-border worker — Citizens — Guichet.lu - Administrative Guide - Luxembourg (public.lu)

For the second case, just write an email to the immigration directorate and they will send you a letter confirming the exemption.

Hope this helps!