I adopted a cat that had been shuffled around from house to house several times in his early kittenhood. One of the people at his last residence was a woman who would pick him up and force him to sit in her lap. "Love me! Love me!" she would say. The cat would just get more and more tense over time. It would have this miserable, trapped expression on its face. Finally, she would let it go, and it would bolt away from her at top speed.
That woman's mistake was in thinking only of her desire for affection from the cat. It wasn't pernicious - she just didn't know any better. She thought that a human picking up a cat should normally result in happy fun petting time, and didn't realize that the cat had its own preferences and might prefer to do its own thing, and have a say over whether or not it was picked up by a creature ten times bigger and more powerful than it. (Does that remind you of an infant human's perspective? It should!)
I also wanted affection from him, of course; he's a beautiful cat. Who wouldn't? I never forced him to accept my affection, but still he shied away from me. If I pet him, he would immediately walk away because it reminded him of all the times before when he'd been forced to accept human affection.
It wasn't until I left him alone and stopped approaching him for affection that he finally opened up to me. I would still try to pet him occasionally, but if he got up and walked away, I immediately withdrew all attention and left him to his own devices. I wouldn't even look at him - I'd just calmly go about my day.
It took awhile - months, in fact. Slowly, tenuously, he became more curious about me. I kept up my strategy, and it worked. It STILL works. Now, he follows me around all the time. I'm the first person he goes to for affection. Sometimes he cries when I'm not around. Once it became clear that he wasn't required to share affection with me, that I would never force the issue, he felt like he had some ability to direct his own actions in my presence.
Your daughter is no cat, but being a mammal, she has a similar limbic system. If she feels crowded, she will associate affection with annoyance and possibly a sense of being trapped. If her father picks her up and she squirms, and he tries to clam her down and coo at her, it won't help matters because she already feels put-upon by the encounter. He can't ease her fears because all she can see is the same undesirable context that has arisen so many times before. She will have to create some positive associations re: being affectionate with her father, and for that to happen, I'm afraid he'll have to wait for her to come to him on her own terms. Let him simply make himself available. Sooner or later, she will relax and come around.
Thomas Paine's experience stands out for me because a key ingredient was changed. His son was used to being triggered by his father's presence around the house. Being in the same place where you've felt cornered or over-cuddled in the past, seeing the father act like he's about to pick him up, will raise the same emotions he felt in the past when that happened, and he will fight against it. Being in the airplane deleted a lot of the environmental variables that would have normally triggered his resistance, leaving him open to react in a more relaxed way. If you want to spur this process, perhaps you could try going out of town. Stay in a hotel she's never been to before, and have him leave her be as much as possible.
One thing I would not do is say to her, "It hurts Daddy's feelings when you won't hug him." That will only give her a complex. She's only responding to emotions she has no understanding of or control over. Adding guilt to the mix will only make it worse. Don't make it about Daddy's feelings. She can't help that. She can't even help her own feelings. This is about her limbic system, and you have to interact with it in a way that it understands. Letting her have a sense of personal agency, of control and self-determination - that's the way to do it!
