The short answer is: "No".

Watching people's eye movements may allow you to follow their patterns of thought - such as whether someone is thinking in pictures, in sounds, in physical/emotional feelings, and so on - but it won't tell you what they are thinking about.

It is also important to remember that whilst many people's eye movements match up with a standard pattern, or model, many other people's eye movements follow quite different patterns.

So, if you want to use someone's eye movements as an indication of internal thinking shifts, it is vital that you start out by asking a few leading questions and calibrate the results (that is, watch carefully to see what happens), before you base your interaction with that person on their "eye accessing cues".
Remember, the eye accessing cues chart was developed as a way of checking whether a client was doing what they had been asked to do, and/or to check what sensory system(s) they used in a particular situation - not for the purposes of mind reading.

For example, if I'm training someone in presentation skills I might ask them to "visualize" being in front of an audience.  In practice, the word "visualize" is not exclusively visual - full "visualization" would involve sight, sound, feelings, smells and tastes; therefore a "partial visualization" may involve any one or more of those systems.  It is up to me to me to watch the trainee's eye movements to see what they actually do when I ask them to "visualize" that scene.  Do they in fact make or recall a mental picture, or do they go for a sound, or feeling, or smell or taste?
And do they stop at just one sensory representation, or do they "stack" them, so to speak?  They might, for instance, look down and right and frown (eye movements are described from the observer's point of view), then up right, then directly to the right, then back to down and right, and end up looking at me with a worried expression.  If I've already established that this person is using the standard pattern of eye movements (see FAQ #9 for complete chart) then this means they had a feeling, recalled an image, recalled some sound, then experienced some fairly negative feeling.
At this point I might hypothesise - purely as a basis for further investigation - that the person was not so much "visualizing" (in the sense of "constructing") but actually remembering how it felt, looked and sound when they were giving a presentation at some time in the past; and that the memory was not a particularly positive one.

Of course I could simply have asked the person: "Do you remember having a bad experience when giving a presentation in the past?"  There are several advantages to checking the person's eye accessing cues over simply asking questions:

• I'm not actually looking for a negative memory; on the contrary, if I get a complete memory, rather than a visualized/imagined image, then I'd rather it was a positive one.  So asking for a negative memory is counterproductive.
• As any psychological researcher knows, people frequently don't give a truthful answer to certain questions.  This may be to protect their self-image, or they are trying to give the answer they think the questioner wants/expects, or because they read a different meaning into the question than the questioner intended, and so on.
  A person's eye movements, on the other hand, tend to be unconscious rather than conscious, and therefore tend to be a more accurate indication of how they are thinking
• By asking the client to carry out some activity I've created the opportunity to see what strategy they use.  In this case:
   
  Feeling   →   Visual recall   →   Auditory recall   →   Feeling
   
  If this turns out to be their regular strategy for visualization then I can help them to speed up and strengthen the process in future by deliberately following this sequence:
   
  Get a feeling   →   Recall a picture   →   Hear the relevant sounds   →   Re-experience that feeling

It must be said that the eye accessing cues are probably the most controversial technique in use in NLP.  Indeed, some observers have noted the substantial number of experiments on the eye accessing cues which have failed to support the standard model.  A few, less well informed observers, have assumed that if these experiments don't actually support the standard model then they must prove that the model is incorrect.  The next FAQ provides an explanation of various reasons why these experiments are likely to fail, and uses one of the best-known (most frequently cited) experiments as an example of the kind of poor design that virtually guarantees a negative result.