Is the red cross a non governmental organization?

Yes, the Red Cross is a Christian organization. The symbolic red cross surrounded by a circle is known as the Mark of Cain. Later Christian organizations such as the Freemasons and the Knights Templar used variations on the symbol and the red cross itself has long been representative of Christian charities.

Officially, organizations such as the American Red Cross and International Red Cross are neutral with regards to religion and politics. That's a difficult sell when your logo is a religious one even if you don't include these organization's decision making which has at times appeared suspicious.

Various controversies over the symbol arise from time to time with some in opposition to the use of clearly religious symbol. See related link for an example.:

Some often confuse the Red Cross symbol with the flag of Switzerland. Such confusion can be easily remedied with a bit of research.