Both the flag of Denmark and the flag of Switzerland feature a white cross on a red background. The Swiss flag is perfectly square while Denmark's flag is rectangular like other countries' flags.
Switzerland's flag hearkens back to the white crosses that confederate troops wore on the clothing in the battle of Laupen of 1339. It wasn't until 1840 that the Swiss troops wore the Swiss flag, a square flag with a red background a white cross in the center. In 1848, the Swiss Constitution established it as the official flag of Switzerland.
The Danish flag, which has a red background and an off-centered cross, is the Dannebrog, which means "flag of the Danes" or "the red flag." The legend of this red flag dates back to a Netherlandish writing of the 1300s, in which the Dannebrog was a heavenly gift.