Dex starts off as being giddy to finally be interacting with Julie as his current north star; he’s trying to mirror her and clearly fascinated in bouncing off of her as she serves as his guide on what he’s aiming to replicate. He falls back into performing as an instinctual reaction to resonate with her, and whenever he is confronted with something that goes "off script" he immediately becomes avoidant and breaks eye contact. He’s taken off guard when Julie tells him he didn’t need the guidance of his parents, because this contradicts almost everything he’s set himself up to believe in. Dex searches for and needs direction and structure, so as the conversation unravels from Julie steering the conversation (her answers increasingly become unexpectant to him), Dex falls into defensiveness and something akin to hostility. The idea of Julie that he has in his head is not equal to the real, fully fleshed out person that she is. He’s seemingly frustrated with the reality that she’s not a symbol to attach to, but an individual with her own contrasting thinking. He starts breaking the "performance" by becoming confrontational and then sets it all off with the dog comment, because he ultimately equates Julie disengaging and leaving their conversation as a rejection of his shown self. It's a threat of a form of abandonment to him, even though she does not know him and his attachment at this point is entirely one sided.