Is it permissible to offer assistance to Muslim countries free of charge through the Red Cross, while not accepting to bear the sign of the cross?

If you are certain that you will not succumb to the pressure of the entity, or that your work will not become a means for them to achieve their false objectives, then there is no harm in working with them in the aforementioned field, with the condition mentioned. The basic principle is the permissibility of dealing with non-Muslims in matters not prohibited by Islamic law. Furthermore, it is better for a Muslim to undertake the distribution of what is to be distributed to Muslims than for a non-Muslim to do so, in order to prevent any evils that a non-Muslim might inject. And the Muslim who undertakes this work should seize the opportunity to invite to Allah the Almighty through word, deed, justice, and fairness.