|This article is about an older version of DF.|
Yes, dwarves do have days off; sooner or later, all dwarves will go on break for some time. There is nothing you can do to stop it from happening, but at least it's less annoying than parties. A dwarf will still go on a break even if they hadn't been doing any work at the time anyway.
Technically, a break is different from eating or drinking or sleeping, but the result is often the same - your dwarves leave their current activity (sometimes in mid-job, dropping any materials where they stand) and walk to where they can take their "break", in whatever form. (See meeting area.)
If the economy is active, then dwarves will wait at least 25 days between breaks, and they will not go on break unless they have earned at least 100☼ during the current season and have enough money to pay their rent and still have 200☼ left over. If the economy is not yet active, then they will wait a full 3 months between breaks and are only 10% as likely to go on break in the first place.
How to stop breaks
You can cancel dwarves' breaks by drafting them into the military and then immediately releasing them. This causes an unhappy thought if the dwarf has no military skills at novice or better. Some players consider this an exploit, while others think it is balanced by the unhappy thought. If all your dwarves have novice in even a single military skill, however, there will be no unhappy thought. You could rationalize this as having "military culture" at your fort if you wanted, but it's definitely an exploit which Toady has acknowledged.
Breaks only last for about two weeks (12.5 days), so unless they cause major problems in your workflow, you can simply wait them out.