It is absolutely correct that he can't kick her out. However, you've been here long enough that you know the answers so well that I'm not sure there's enough detail to help the newbie.
Basically, it's the marital home. Both of you are entitled to live there until the court rules otherwise. That would routinely happen at the time the divorce is finalized - with one person getting the house and the other leaving. However, it is possible to ask the court to assign the house to one person right from the beginning of the divorce process rather than waiting until it's final. You would file that as a request at the time of filing for divorce or shortly thereafter (if he files for divorce, you need to prepare a response and you would request it then). You would also file for temporary custody orders at the same time. If the home is in his name and was his before you married, there's almost no chance that you'll get it even on a temporary basis. Similarly, if it's in your name, he won't get it.
If the house is in both of your names, he court will consider a number of things in determining who gets temporary possession of the house. Kids are a big one. The person who has been the primary caretaker for the kids has an edge when it comes to physical custody (if you can't agree on 50:50 or some other split).
If the house is in both of your names, the court would prefer not to move the kids out of the marital home, so the person who gets temporary physical custody of the kids would have an advantage when it comes to the home. HOWEVER, it's not a slam dunk - you need to be able to afford to keep up the home (either on your own or with temporary support from your stbx). While it is unlikely that you would get support after the divorce, you MIGHT get a small amount of support while the divorce is pending. Certainly enough to support the kids, but probably not enough to pay for the house, kids, yourself, and so on.
So, you will need to get as good a job as you can - as soon as you can. First, you'll need it once the divorce is final if you don't want to starve. Second, if you have enough income to pay for the house, getting temporary possession is more likely.
Once the FINAL decision is made on the house, the person who keeps the house will have to reimburse the other person for their 1/2 of marital equity (basically, the amount that home equity increased during the marriage) and both will have to execute the appropriate documents to get the home out of the others' name.