I agree with the other ladies, if you punish him for not eating no wonder he is upset and throwing-up. He is associating eating with punishment. Try some basic foods like RockRose stated and don't give him the fries. Like they say, when they are hungry, they will eat. Maybe having mom walk out of the room while he eats may help with his anxieties. Best of luck to them.
Sounds more like food aversion to me--this isn't usually about the food but about power relations/control with the parents/feeders. I knew a kid who would gag and puke for anything but pasta/sauce or french fries. This happened from age 5 to maybe 11--there was a LOT going on in the home (new siblings, mentally ill mom moving away, new step, trouble with dyslexia, etc). I would think a therapist might be a good idea if you decide (or a gastroenterologist tells you it's not a "food issue")--.
You can also of course trick them (make french fries out of something else (tofu? chicken?), force them (very hard to do) or wait for them to get hungry with a plate of food always out for them (it's not like they can go out to store and buy their own supply). I don't really think tough love works on this since the power struggle just feeds the flame. sorry you're going through this...
I agree with Rock Rose, give him simple foods that will not stimulate the gag reflex.
And practice a little tough love. Get rid of the junk. He may be resistant at first, but he will soon eat.
I personally think a lot of picky eaters are just not exposed to enough good food choices from early on. I have two kids who eat just about everything. My dd hates potatoes and my ds hates spinache. They ate regular meals with the family from a very young age. Keep offering the healthy choices and not give in when he screams for junk, and I think he will catch on.
He sounds like he has sensory defensiveness issues. She needs to find foods that don't gag him, because literally, that's what foods do to children with sensory defensiveness.
A lot of vegetable and fruit juices, WITHOUT CHUNKS, and browned hamburger meat, and angel hair pasta with butter, and probably chicken nuggets dipped in ranch dressing might be good. Crispy granola bars might be good, too.
Punishing children who gag when they eat foods isn't the right way to go. He needs to find a variety of foods (probably a limited variety) that he will eat, and otherwise, give him multi vitamins.