The lawyer was not right, if that is how he said it. It is only true that your boyfriend may be eligible for cancellation of removal for having been here for 10 years. Cancellation of removal is a form of relief when a person is put into deportation proceedings. If your boyfriend gets caught for being here illegally then he will be put into deportation proceedings. Because he has been here for 10 years or more he will be able to stay and receive a greencard only if he can show that his U.S. born children will suffere extreme and unusual hardship. This is not easy to show and this is something to try only if he is put into deportation proceedings not soemthing he can apply for right now.
The medicaid issue will likely effect you more than him. You may be committing fraud, a very serious offense. I would have your boyfriend speak with an immigration lawyer, just incase there are more options for him, such as marriage to a U.S. Citizen (may or may not fix the situation depending on many variables) etc... There are many issues here. Your boyfriend would benefit with a full consultation with an experienced immigration lawyer, there are probably a few in Dover.
I think you should also speak with a Criminal Lawyer to find out what trouble you may be getting your self in because of the Mdicaid issue.
No this lawyer is absolutely wrong. There are some unethical people out there, some are lawyers, others are posing as lawyers to get your money. Be careful. Your husband may be eligible for cancellation of removal but its very challenging to establish relief because it requires "exceptional & extremely unsual hardship" to you & your 2 kids. There may be other forms of relief such as 245(i) if he is eligible, but you need to speak to a lawyer. Medicaid fraud is a hugely punishable offense. How is he getting checks at his job in your name? Is he helping you with this scheme? If so, it can be a huge problem for him too. Its usually a State Jail felony if caught & that could potentially knock him of relief.