[Mythotical]

Yes if they save the option no matter what computer they login from it will stay whatever they choose until they change it.

Ok follow these instructions:

Go to AdminCP > User Profile Fields > Add New User Profile Field > Single-Selection Menu (from drop down) > click Continue

Now on the next page:
1. Title - Give it an explanatory title in only a few characters such as: Turn On/Off Shoutbox
2. Description - Give a small description to explain what the option is for.
3. Leave "Profile Field Category" alone
4. Options: Enter the follow

Yes
No

5. Set Default: This I would leave as is.
6. Display Order: Leave this option alone
7. Field Required: Select whether the field is required to be changed when the person registers, updates their profile, etc. Or just leave as No
8. Field Editable by User: Leave this option alone
9. Private Field: Choose whether others can see if they have turned off shoutbox or not
10. Field Searchable on Member List: Set to 'No'
11. Show on Members List: set to 'No'
12. Option Input: Leave this entire area alone
13. Last box, choose what UserCP page to place the field on, probably leave as-is.

Click save and your done with that one part. Remember to take note of the Profile Field ID.

Now I'm not sure what shoutbox you have installed so all I can say is try to find where the code is for $shoutbox and then put this around it:

<if condition="$field[X] == '0'">
$shoutbox
</if>

Let me know if you need further assistance such as locating the shoutbox code.