Do only humans spend a lot of time daydreaming (or having "stimulus-independent thoughts")? A paper 
Killingsworth et al., A Wandering Mind Is an Unhappy Mind, Science 12 November 2010: 932 (or a free pdf) starts with the following statement:


  Unlike other animals, human beings spend
  a lot  of  time  thinking about  what  is  not
  going on around them, contemplating
  events that happened in the past, might happen
  in the future, or will never happen at all. Indeed,
  “stimulus-independent thought” or “mind wandering” appears to be the brain’s default  mode
  of operation (1-3).


While the claim is supported for humans by the cited papers and their finding:


  Mind wandering
  occurred in 46.9% of the samples and in at least
  30% of the samples taken during every activity
  except making love.


I couldn't find a strong evidence that for all other animals it is not true. Even if it, arguably, may sound plausible for domestic animals, it seems less convincing for primates, elephants or dolphins (without a proper experimental support).

Are there any research on "wandering mind" for other animals? If so, do they support or refute the "unlike other animals" line?
It seems that it is still a matter of debate whether animals are capable of mind wandering. For instance, there are a lot of publications about foresight, a future directed instance of mind wandering. Much of it comes from one group, e.g., 


  Suddendorf T, Corballis MC. (2007) The evolution of foresight: What is
  mental time travel, and is it unique to humans? Behav Brain Sci. 
  30(3):299-313; discussion 313-51.
  
  Suddendorf T, Corballis MC. (2010) Behavioural evidence for mental
  time travel in nonhuman animals. Behav Brain Res. 215(2):292-8.


But, for a review see: 


  Cheke Lucy G., Clayton Nicola S.. Mental time travel in animals. WIREs
  Cogn Sci 2010, 1: 915-930.


Or for a quite current example, see: 


  Osvath M, Karvonen E. (2012;7) Spontaneous innovation for future
  deception in a male chimpanzee. PLoS One (5):e36782.