Izuku Midoriya and Mirio Togata's choice to permit Eri to return with Overhaul following their initial encounter was one of the pivotal points of the "Shie Hassaikai" storyline of My Hero Academia Season 4. Both heroes-in-training were haunted by their decision to allow a scared young girl to return with a known villain for the rest of the arc, especially after discovering what Eri was being used for, even though neither character at this point understood the extent of trauma and suffering Eri was going through.
That makes one wonder what could have happened if Deku had been more adamant about preventing Eri from going back to Overhaul. Through the "Shie Hassaikai" storyline, saving her became his and Mirio's main motivation to make up for not protecting her. How would Overhaul have responded if they had made the decision to keep her with them at any costs? What would this have meant for the rest of My Hero Academia Season 4?
The First Time Eri Met Her Future MHA Heroes
Deku and Mirio first meet Eri in My Hero Academia doing a routine patrol for Sir Nighteye's Agency, when the frightened girl stumbles into them running out of an alleyway. She is followed out of the shadows by Kai Chisaki, also known as Overhaul, leader of the Shie Hassaikai mob organization and the target of Sir Nighteye's investigation.
While Mirio attempts to quickly withdraw from the situation to avoid drawing suspicion, Deku notices her trembling and bandaged arms and inquires about her injuries. Overhaul claims Eri is his daughter, her fear comes from a recent scolding she received, and her injuries were caused by her clumsiness. Despite Mirio's subtle urging to leave, Deku's heart is swayed by Eri who begs him not to leave, and demands Overhaul better explain the situation.
Overhaul relents, and invites the two heroes into the alley with him so he can clear up their misunderstandings. Once inside, he casually begins to remove his gloves; in response, Eri quickly runs back to him out of fear that he is preparing to use violence. Deku and Mirio regroup with and relay the encounter to Sir Nighteye, who reprimands Deku for not thinking with his head and potentially compromising the target. When both Deku and Mirio later learn that Eri's body is being disassembled by Overhaul for the manufacture of Quirk-erasing bullets built from her Rewind Quirk, they both feel extraordinary guilt and desire to right their failure to protect her.
What if Deku and Mirio had insisted on taking Eri? What would Overhaul have done?
Considering Overhaul was in the process of removing his glove, it's likely he was planning to kill Deku and Mirio on the spot. During the later raid on the Shie Hassaikai compound, Mirio singlehandedly put Overhaul on his last legs while also fighting other members of the Shie Hassaikai and protecting Eri. While neither Deku nor Mirio knew what Overhaul's Quirk was at this point, it's still likely that Mirio would be skilled enough to defeat Overhaul in a pure fight if the latter was planning to react violently. If Deku attempted to assist, his lack of experience would likely hinder more than help, and he would be better off escaping with Eri and finding backup from other nearby heroes.
However, considering the destructive nature of Overhaul's Quirk and the fact that civilians were around, if Overhaul was intent on fighting he could've caused many causalities and collateral damage by molding the ground and buildings around the area -- which neither Mirio nor Deku would've been able to prevent. Furthermore, while they could claim suspicions of child abuse, the two heroes wouldn't yet have any legal justification or evidence they could use to prove Overhaul was a threat that needed to be defeated.
Without any clear reasoning to use for being able to beat down Overhaul in the middle of the street, even if Mirio and Deku were to survive they would have no grounds to capture him and would likely be punished by the law for hurting what appeared to be an innocent man. It's even possible that the police would force Deku and Mirio to give Eri back to her rightful legal audience since there was no clear evidence of wrongdoing, making the rescue attempt pointless.
How the Shie Hassaikai arc in MHA might have developed differently
Rescuing Eri early would cause a butterfly effect on MHA that can't easily be measured. However, two things are clear: Sir Nighteye's Shie Hassaikai raid would've never happened, and Overhaul would try to get Eri back at all costs. Sir Nighteye already disapproved of Deku's decision to hold on to Eri, saying his arrogant desire to save one girl could've ruined an operation to catch a villain capable of far greater harm. If Overhaul murdered civilians in his attempt to get Eri back, he and the Shie Hassaikai would've gone deep into hiding -- meaning Nighteye would lose his primary target and be forced to end his investigation due to a lack of information. Deku would likely be fired from the Nighteye agency, at least until results came back that Overhaul's Quirk-erasing bullets were made out of Eri's body.
In the event that Overhaul was apprehended and later freed by the authorities, he would do anything to recover the main component of his grand scheme. He would abandon all pretenses and employ all available means to launch an attack on the location where Eri was being held in order to free her, leaving a trail of carnage in his wake. He may have even taken Shigaraki and the League of Villains up on their offer and mounted a counter-raid on the Nighteye agency, given how eager they were to work with him.
Overhaul's vengeful nature would lead him to find out exactly which heroes took Eri away from him, so Mirio and Deku would also become targets. Depending on how quickly Overhaul's plan was discovered -- and how long the heroes would have to prepare -- it's possible the villain's assault would cost numerous lives, including Sir Nighteye, who would be at the center of the conflict.