That's not even remotely true.
First and foremost, the shemagh originates from a "hot ass desert". They're a traditional headdress worn by various cultures in the Middle East and are used primarily for protection from the sun and the sand/dust.
Secondly, here's a photo of the Navy SEALs prior to Operation Red Wings (where most of those in this picture were killed, may they rest in peace). You can see Navy SEALs Marcus Luttrell, third from right, and Eric S. Patton (KIA), second from right, wearing scarfs in Afghanistan.
More relevant to us, here is a MARSOC CSO wearing a shemagh while deployed (country not specificied, but looks like Afghanistan).