This lawsuit is not about J&J trying to prevent the Red Cross from using a red cross as their symbol for their charitable relief efforts. J&J and the Red Cross have gotten along for decades each using the symbol. However the Red Cross has only been able to use the symbol for charitable, not for profit purposes.
The problem that has developed recently is that the Red Cross has started licensing the symbol to other companies for use on their products that are being sold for a profit and competing with J&J's products.
The Red Cross is the bad guy in this situation IMO. They are the ones that violated the 90+ year old agreement over the symbol...and for the record, J&J offered to go to third party private mediation to try and resolve this dispute rather than file the lawsuit against the Red Cross but the Red Cross refused.