I recently received a new gadget: The Jawbone UP band. Look it up on Google, but the basic idea is it is a bracelet you wear 24/7 that measures all kind of things while you move around.
Should it be removed on Shabbat?
TL;DR: Please do CYLOR, but one should probably remove it. I would not try to argue that it is clearly forbidden or not, though I would assert that the issues below make this complex enough that one should avoid it until they can consult with a competent halachic authority.
There are three major issues, that I can see, with wearing one of these bands, like the Jawbone UP or the Fitbit Flex: Carrying outdoors, Measuring, and Electricity (as @trying mentioned, Writing is often discussed too). Each of these things are generally forbidden on Shabbos, with some exceptions.
In order to allow wearing one of these outdoors, one would have to argue that they are considered a "garment," (or for women, an "adornment"). This is not insurmountable, and one could argue that it's similar to a watch, which many authorities allow. CYLOR
The sages forbade any acts of measuring (where not essential for health or food), on Shabbos. These bands measure and record one's physical activity, and could certainly run up against this issue. CYLOR
Writing (because why not?)
Could be an issue, although the fact there is only an electrical recording, and nothing visible (even on a microscopic scale), may make this a non-issue. Then again, it may not. CYLOR.
Essentially, these activity wristbands contain electrical sensors to measure things like movement and vibration. They then record the measurements, again, electrically. CYLOR
There are some factors that may allow one to be lenient in specific cases. For example, indirect usage can be used as a mitigating factor. @trying gives the example of opening a refrigerator, which could indirectly cause the compressor to turn on by lowering the temperature. Some authorities allow this, usually provided the effect is not immediate. It seems to me that activity bands cannot make use of this leniency, since there is no indirectness in one's activation of the sensors. They directly sense and record any movement by the wearer.
Another possibility alluded to by @trying, is the idea of "psik reisha," unintended results that were unavoidably caused by one's actions. Cases where the unintended result is also undesired are called "psik reisha d'lo nicha lei," and can be combined with other mitigating factors to allow certain activities. This is less clear, and I can see some arguing whether walking/moving/sleeping while wearing these bands would be activating the sensors "unintentionally." That seems unlikely to me: one is wearing the band for the express purpose of recording movement! Additionally, it would be difficult to argue that the resulting measurements are "lo nicha" (unwanted).
R. Gil Student here quotes R Yisrael Rozen of Tzomet (an Israeli institute dedicated to seamlessly merging Halachic Judaism with Modern Life) regarding a Fitbit with all displays turned off during Shabbat. R Rozen rules that
if you normally wear the FitBit so that it has nothing changing on its screen due to human movement and no alarms or alerts, then technically you may wear it on Shabbos. You are wearing it the normal way and it does not do anything observable nor cause anything you desire to occur. He compares this to using water that is measured by a water meter. When using the water, you are not directly doing anything forbidden nor causing anything observable to occur.
However, Rav Rozen adds that as a matter of public policy, people should avoid these types of things absent a serious medical need. There is a slippery slope of using electronic devices improperly. This device can also be considered a debasement of Shabbos or an interference with the Shabbos atmosphere. He adds that you certainly should not use such a device to wake you up on Shabbos.
Secondly: I see two possible questions:
(1) is it a problem to do something on shabbos which will trigger an electronic device?
(2) is this type of writing to the hardware considered "kosev"?
Here are some sources which may shed some light on this topic.
though most Shabbat observant Jews permit opening and closing a refrigerator during Shabbat, some authorities require that the door only be opened when the refrigerator motor is already running. Otherwise, the motor will be caused to go on sooner by the increase in temperature indirectly caused by the flow of heat from the outside. Most refrigerators and freezers automatically set the motor to turn on and blow cold air whenever the thermometer registers a temperature that is too high to keep the food cold. However, Auerbach and most authorities permit opening the door because this result is indirect, and because there are additional grounds to be lenient.
While most rabbis have ruled that the example of intentionally letting cold air into the room to operate the thermostat constitutes a violation of Shabbat, if the person opens the window for some other, legitimate, reason, and the cold air enters as a side effect, no violation has occurred. Additionally, most agree that if a person who has no intention to operate the thermostat does something which happens to operate it, no violation has occurred.
However regarding the issue of writing, all I can say is Making a mark (even with a fingernail) for a significant purpose is considered kosev midrabanan. but I'm not a rav, so I won't try to make something up.