Poor man's Alcubierre drive? In general relativity,  the fundamental notions are the curvature (Ricci tensor) and the stress energy tensor. Energy density and curvature are connected . 
The Casimir vacuum between parallel plates could fulfill the negative energy density requirements (positive energy density for Casimir spherical shells as far as I know  , but I don't have strong references in this direction). 
In vacuum,  in the absence of any other forces or fields,  does matter tend to follow the gradient of the vacuum energy density  (  for example  from high vacuum energy density  to low vacuum energy density )?
In vacuum,  in the absence of any other forces,  a ship with a large number of Casimir plates at one end (I won't bother to put spherical shells at the other end), will it tend to move without the need of any fuel (as slow as it may)?
A picture's worth a thousand words. View here .
In this design the Casimir plates are parallel to the direction of travel. A similar design could be considered,  where the Casimir plates are orthogonal to the direction of travel.The ship moves for the same reason the plates are pushed together in the original Casimir experiment,  following the gradient of the vacuum energy density.
If not, please explain. 
 A: 
In vacuum,  in the absence of any other forces or fields,  does matter tend to follow the gradient of the vacuum energy density ( for example from high vacuum energy density to low vacuum energy density )?

I understand your design and the crux of the question is the one above.
In this link, pressure as energy density is discussed.

Pressure in a fluid may be considered to be a measure of energy per unit volume or energy density. For a force exerted on a fluid, this can be seen from the definition of pressure:

$$P=\frac{Force}{Area}=\frac{F}{A}=\frac{F\cdot d}{A\cdot d}=\frac{W}{V}=\frac{Energy}{Volume}$$
If one considers the energy density of space as a fluid then it seems there would be a pressure difference that could create a propulsion. So it would be the pressure, on the matter on one side, to lack of pressure on the Casimir side. Too many ifs in the process though. It would not be a complicated experiment though.
A: https://www.google.com/url?sa=t&source=web&rct=j&url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110015936.pdf&ved=2ahUKEwi00rfH4uHkAhVCyaYKHXJECb8QFjAAegQIAxAB&usg=AOvVaw214vpEClifvcvPQCaBLUTx
This is from NASA website on advanced propulsion research. They are playing around with this micro warp fields to try to measure something tangible. Point is even if the force is small it's still great for space travel cause you have a long way to accelerate and there is no drag. But this is still based on GR ,  and GR  is just a math model, granted with decent predictive power and insight into some deeper truths , but it's far from the best picture of reality. People online can only try to tackle this problem combining GR and qft but that just doesn't work ,cause they are just some empirical math models used for their respective purposes and based on that they can neither refute nor confirm your idea. I saw your drawing and yes it's an interesting point creating lower density of vacuum on one side and somehow keeping the plates from closing down so that lower density persists . But I will bet you anything this kind of physics is classified. It has nothing to do with typical politics ,left or right, I don't care about that.i haven't made the experiment nor have the theoretical knowledge as no other physicist has to talk about applications of quantum vacuum
A: No it wouldn't. It's like you being in space pulling the the earth with large rope you will move towards the earth much more that the earth will move to you, in the end momentum will be conserved , so no net movement in either direction.
That being said NOBODY can tell you that warp drive or even pulling momentum and energy from " nothing" ( vacuum)  is impossible. QFT is based on flat space so that means conserved momentum and energy is basically an axiom, GR requires negative energy for propulsionless drives which hasn't been found yet which doesn't mean it won't ever be found. Both theories fundamentally being incomplete there is always a possibility . But you have to go back to reality. Even if warp drive turns out to be possible or even operational, you'll be the last to know about it, since in this world money is first priority. Given a long history of scientist taking part in cover-up in the interest of industry, any threat, like the possibility of propulsion not based on Newton's 3rd law, to the multi trillion dollars petrol industry will be utterly dismissed .
