What are the most common mice in New England?

The deer mouse and the white-footed mouse.  

What do they eat?

Anything they can get hold of!  It is important to note that most mice come inside to over-winter with us.  To them, our homes are just hollow trees.  They can get all the food they need from the "natural" environment and store it in our warm houses.

Do they really carry and spread disease? 

Food poisoning (from Salmonella primarily) from droppings is probably the greatest threat.  Hanta virus can also be a big one - but primarily in drier environments.  Leptospirosis occurs when a cut or wound is contaminated by mouse urine.  Allergies are huge.   Mouse urinary proteins can trigger severe asthma in children and others.  Mice have minute (some say none at all) bladders and pee thousands of times a day, called microdroplets.  So, they are not good critters to live with!   It is also estimated that over 25% of all (house) fires are caused by mice stripping insulation from wiring.