I agree 100% with cbg. And remember this, the unemployment system is NOT your enemy. They have seen both sides of these situations for a long long time. And they are very familiar with employers who lie to employees and misinform them about unemployment totally. If it were up to the employer to make the decisions, NOBODY would ever get approved to draw benefits, because they have to pay more if you do.
And the unemployment system works under the assumption that either you OR the employer may be lying. They're not going to give them more credence than you simply because they are the employer.
You have your story. They have their version of what happened. What will happen in the hearing is that either they, or you, will speak first. If it's them, they will tell the circumstances under which they either terminated you, or as they claim, you quit. You don't say anything while they are talking. If it is you, you speak first and then they will speak. No interruptions, questions, objections, etc. Then the hearing officer will possibly ask both of you, "Do you have any questions for this other party?" You can question them, and they can question you at this point. You answer any question they ask you honestly and clearly. You ask them about anything they've said that was questionable to you.
When you speak, what you do is give a summary of exactly what circumstances led up to your being terminated. Be very definite in stating that you did not quit your job. If you have emails that show this, practice working them into your talk. They'll want copies of these distributed to each person present at the hearing. The hearing officer will take care of this.
Whatever you do, do not write down everything you are going to say and go in and read it off. But do make a little summary of what you will say when they say, (the same questions you were probably asked in the fact finding phase) and practice speaking your part to answer these questions. What happened? Why are you no longer working for this employer?
While you are speaking your former employer's representative will not be able to cross examine you or argue with you. The appeals referee may interrupt or guide the story if they feel you are wandering off topic or if there is something that you haven't covered that they want to know about.
While MA is a great state to be a claimant in, and they are particularly known to be sympathetic to non job related issues, do stay away from a lot of discussion about the daycare places being "too expensive" for you to work full time and put your child in. This could wander into the area of whether or not you are able and available for work right now. That's a personal issue that doesn't need to come up. Unemployment insurance isn't based on need or how poor you are. You could have a very wealthy husband, or a trust fund, and money to pay daycare would never be an issue and you still weren't able to find a good daycare for a newborn on such short notice.
But seriously, write down what you are going to say and get up and practice it. Make very tiny talking points on your hand or a tiny post it note or something to remind yourself. Practice is your friend, just as if you were going to do public speaking. But this isn't public speaking, really. It will be a small hearing, just two or three people present, and you do not have get up and be F. Lee Bailey (gee, I don't know, who's a famous attorney I can use as an example?) But anyhow, they've done thousands of these hearings. It is set up so you do not need to hire an attorney to represent you as long as you are able to speak coherently and reasonably. There is no chance that an attorney will do anything except get you what you might've gotten anyway, and then you'd have to give part of the money to them for doing something you can very well do for yourself, which is tell your story quietly, calmly, briefly and stress the things we have told you to.
The way it happened is the way it happened. Being consistent will be to your favor. Being non argumentative is also good. You don't have to deny or refute everything they say or try to claim about you.
Try to relax and prepare. Be sure you are making the weekly certifications for benefits correctly and are fully registered for and seeking other employment as required. The employer is always allowed to do this second appeal. But you have won the case once and you do not need to change anything, because they are not going to have any new evidence that you quit your job when you didn't. They'll just want to reemphasize something that didn't win before. Good luck to you .