A clarification

The Red Cross. It's symbol is... wait for it... a red cross. For example:

This has now come to symbolise First Aid in general as well. (Although this is, traditionally, supposed to be a white cross on a green ground).

The Swiss Flag is the opposite of this (or, indeed, vice versa as the case may be). A white flag on a red ground. Like so:

If your first aid kit looks like this, it is, quite simply, wrong: