You mean they think they're suffering from the effects of WiFi.
I have yet to see an EHS sufferer who's not a crackpot. I suffer the same symptoms on a regular basis, but I know that the headaches are from my sinuses, the skin rash is from the dry air, and the fatigue is from not getting enough sleep.
I haven't seen any rigorous scientific studies that support EHS as a legitimate thing. In fact, some of the people "suffering" from the syndrome complain about the effects even when the nearby transmitters are turned off or when they're near a receiving antenna.
And do any of these people own microwave ovens? My microwave leaks 20-30 times more energy while cooking than my WiFi gear radiates on purpose. (WiFi is limited to something like 100 milliwatts. A microwave can leak 5mw per square centimeter.)
So is WiFi going to harm a baby? Considering the fact that the FCC has a century of well-documented radio science, the answer is no.
As to your friend... does he wear the hat in the house, too? Radio waves don't stop at your front door. I'm sorry, but he's not suffering from EHS. If he gets regular headaches, It's far more likely he's simply got sinus problems, mild diabetes, or a thousand other things that have exactly the same symptoms as EHS "sufferers" describe.