Since they are veering from the book, I have a different theory as to June's reaction on watching Daisy.
I think June planned to recruit Daisy much earlier on, and was hanging out in her parents' shop to do just that. She recognised Daisy as a livewire, headstrong girl and thought that she would be a good candidate for MayDay. Also being a similar age to Hannah would have meant that she could find a way to get her directly to a placement in Hannah's school.
I think June orchestrated the attack on the shop in order to radicalise Daisy and give her a reason to join MayDay. (Either directly and making it look like Gilead, or indirectly by feeding fake intelligence to Gilead about Daisy's parents).
June has always put other people in danger to try to reunite her family, and I think Daisy is just another cog to her. It was all too slick for her to turn up after the attack and whisk Daisy away without pre-planning.
Also when Daisy asked about June on the bus, the Guardian's reaction made it clear that June hadn't gone through "official" MayDay channels to get Daisy into position as a Pearl Girl.
TLDR: June groomed Daisy's family and orchestrated the attack on her parents to radicalise her into joining Gilead.