Welcome to Gayspeak Mac
Well Im no expert on this subject, but I can give you some advice from my limited experience with my mom.
First off the ESTA gives him absolutely no legal right to remain in the USA past the 90 day stamp in the passport. Its just designed to allow for a smoother passage through US border controls at the main entry points, as many times as you want within the two year valid date.
My mom (British) married her current husband (US Citizen) two years ago. They are both seniors.
The wedding took place in the UK, and three months later they both returned to the USA where she started the "green card" process. To cut a long story short, she overstayed the 90 days by 3 days (she had been ill so couldn't travel on their planned date back to the UK) They both returned to the UK a few days later, and thought nothing of it. There was no drama at US border control on exit etc.
They were in the UK for two months, booked their return flights etc and headed back to Boston. ESTA still valid for my mom. Simple right? WRONG big time. She arrived in Boston presented her passport at US Border control, was immediately arrested, detained for 6 hours and deported back to the UK the same day as an illegal alien who had previously overstayed her 90 day visa (Remember this is an 80 year old senior, already married to her US husband for 8 months at this point, and the green card process for her as a spouse had already been started) Husband was allowed entry. He immediately came straight back to the UK on the next available flight. Its taken them a YEAR of form filling, US Embassy and State Department involvement (and the intervention of his Congressman) for her to be allowed back into the US and restart the green card process.
So my message is very clear here. If your bf has already exceeded his 90 day stay in the US he will be classed as an illegal, regardless of any relationship he has with you. If he then leaves the continental USA, he will be denied re-entry.
The good news is, if he has not exceeded the 90 days, then you need to arrange for him to leave - even if for a day, and re-enter the USA through a recognised border crossing and have his passport re-stamped with a new 90 day visa. In the meantime you can start off the green card process. But be warned, its an extremely complicated process and I strongly advise you to engage some legal assistance in the matter.
Good luck