It can be done but it should not be done that way.
When you inherited the property from your mother, you got what's call a "step-up in cost basis" on the house. That means that the base value of your home was increased automatically to the fair market value of the home at the date of your mom's death. This is good because if you were ever to sell the home, you would pay less tax on it.
Now, if you do the same thing and leave the house as an inheritance for your children, they would against get the step up in cost basis and they would not have to pay tax (or pay less).
In addition, suppose you quitclaim the house to your kids and they get into a car accident and they get sued. Guess where they're going to go for the money? Your house.
Or suppose that they get married and subsequently divorced, guess where their spouse will go after for their share of the marital estate? You guessed it, your house.
So the bottom line is that you should work with an estate planning attorney to plan out what you should do in terms of taxes and wills. Do not go ahead and start transferring them around because you believe it to be a good idea. It might have unintended consequences.
I agree with Attorney Cheong, and emphasis his recommendation that you speak with a competent estate planning attorney. I'm also wondering about your husband's request that you give him a share of the house. Are you separated or contemplating a divorce? If so, regardless of the fact that you inherited the house, it will be considered in a valuation of marital property should you divorce.
Even if you are not contemplating divorce, you should also consider your long-term care needs. Today's laws regarding qualification for state aid for long-term care are very stringent, and giving away your home can have serious consequences if you do need to seek state funding for your care.
Those are two more reasons why you should absolutely not simply deed away your interest in your home. It will be well worth your while to work with a competent attorney.
As I am not licensed in your state, I can only provide you with general advice. I would suggest that before you make any type of conveyance, you seek legal counsel with the attorney who you intend to have draft your will. He can advise you as to any consequences you may incur, such as tax consequences, should you make the conveyance in the way you are talking about. Once you know all the possible consequences of making conveyances in different ways or forms, then you can make an informed decision as to what to do.