I haven't seen the Larry King interview and to be honest I haven't been following this case closely, so please do correct me if I have missed out on some important fact.
My understanding is that Mayumi Heene *is* a legal immigrant to the US, and that she holds legal Permanent Resident status based on her marriage to a US citizen. Further, because she has been in the US for so long, she would have been eligible for US citizenship a long time ago. However, Japan is one of the few countries that does not allow adults to hold dual citizenship. (For children it is OK until age 21, then they must decide which country they want to commit to for the rest of their lives, which some people consider to be a bit harsh.) So for Mayumi, as for most Japanese immigrants to the US, US citizenship is not a viable option if she wants to retain the right to move back to Japan (with or without her family), for example in case she needs to care for aging parents, or for any other reason such as becoming widowed or divorced or just missing good Japanese food. Instead, she needs to be a US Permanent Resident forever. How is this different from being a citizen? The main difference is that she cannot vote, or run for public office. Other than that, she has the same rights and responsibilities as an American citizen (paying taxes, etc).
But there is one problem! "Permanent Resident" status is NOT in fact permanent. It can be lost or taken away at any time (unlike citizenship, which is generally only taken away in a case of treason or something like that, and only very rarely). And there is a relatively recent law (within the last 20 years, I think) which says that a non-citizen convicted of a felony should be deported. I remember seeing a case on TV some years ago about a woman, I think she was Irish, being issued with a deportation order because she had got in a fight with another woman and had been charged with assault, which is apparently a felony. She had been living in the US since childhood, but had never bothered to become a citizen. Big mistake on her part, but there you go.
So the issue is not that Mayumi is an illegal immigrant, because she isn't, the issue is that as a legal immigrant she can still be deported, or threatened with deportation. Hence in the plea deal it was prudent for her husband to plead guilty in exchange for a promise that she would not be charged with anything. In the worst case, she could have been deported, and their family torn apart. It would be difficult to imagine the Japanese government issuing her husband with a spouse visa, if he had a felony conviction. (But they would probably let her bring her children back to Japan, since they are not felons.)
Obviously, it would have been better for them not to get into this situation in the first place. This is a cautionary tale for any bi-national family...stay out of trouble because the consequences could be far worse than you realize. Seriously, if someone hits you, whatever you do don't hit back, because you could end up being deported. And if your country allows dual citizenship, do at as soon as you are eligible, no matter how much it costs and how onerous the paperwork may be.