@Lila,
Florida is an Equitable Distribution state. So unlike a community property state which just takes everything and divides it down the middle no matter who owns it, an equitable distribution state says they divide it "fairly." Thus some property, like inheritances, can be separate property, and marital property is all the stuff and money you earned during the marriage. He gets his separate propert, she gets her separate property, and they take the marital property and spit it 50/50. That's SUPPPOSED to be how it works.
Since the home is an inheritance, in her name only, with no mortgage, she will definitely have a claim to say it is separate property...BUT (and this is a big but) it also sounds like she commingled it while they were married. That is to say, if a separate property is remodeled using joint money, or maintained using joint money, if they used it for vacation or lived in it--then usually they consider some portion of it to be marital property. At minimum he'd be due his half of what marital money was used to maintain the place. Thus she might be due 75% and he'd be due 25% but it sounds to me like he'd have a claim anyway.
Now Florida does not consider abandonment, so either one of them can leave without raising the issue of abandonment. If one or the other were to leave, it might affect "primary care giver" of the children, which might affact child support. However, even if his name is not on the title nor on the deed nor on a mortgage, he has been living there and thus it would be considered his residence at least. I don't see any legal right for her to kick him out.
Thus I would recommend one of two things. Either 1) reach some civil marital agreement about who is going to live where and write it down so that both party's interests are protected--use lawyers to write this and file it. Or 2) go to court and ask the judge for a temporary exclusive rights/use of home order. Essentially this is the judge saying "Okay until we can go to trial, HE will live here, SHE will live there, and they cannot bother each other." Then if he does try to come bother her, she has the law to back her up.
I would NOT recommend trying to evict a spouse. He's not a tenant (unless they have some sort of written lease agreement) and thus it would essentially make her look bad.