How do I backup just a single user account with OS 10.10 Yosemite?
Open Disk Utility application.
From the Menu Bar select "File→New→Disk Image from Folder"
Select home folder of the user account. Click "Image" button.
In "New Image from Folder" window enter:
Save As: Name for .dmg file that will be created.
Tags: Leave blank.
Where: Where to place .dmg file.
Image Format: compressed
Encryption: none
Click "Save" button.
In the "Disk Utility" window, highlight the new .dmg file. From the
Menu Bar select "Images→Scan Image for Restore".



  You now can mount the new .dmg file and access the saved files. You may have to use the chown command to reassign ownership of any files copied from the mounted .dmg file.


Update 1:

Personally, I would preform a backup of a users files while booted to a recovery partition or another partition containing OS X. Using internet recovery would also be preferable. 

If you choose to preform a backup while logged into an administrator account, then choose to save the .dmg file to location where the user would normally have permission to save such a file.

For example, say I (davidanderson) open the disk utility and try to backup ronjanzen's files to the default folder /Users, as shown below.



I would expect the following error message, because I do not have permission to create the file ronjanzen.dmg in the folder /Users.



Now, if I choose to save to my Documents folder, then I will succeed.



The other alternative would be to give read/write permission to the /Users folder. If you do a "Get Info" from the Finder on folder /Users, you get the following.



If you modify "Sharing &amp; Permission" as shown below, you will not get the above error message.