I had my Nikon D7000 on single-point auto focus and it just would not focus on the spots I wanted. I have marked the spots that could not be focused on with red circles, and the parts that could be focused on with green circles.
I was using a Nikon 35 mm f/1.8, although I don't think this was a lens' problem since it has happened before with other lenses too.
Short answer: Current autofocus systems only work when the AF area contains high contrast. The places where it doesn't work don't contain enough, and the areas which do work, do.
Here's what's going on in more detail:
There are two different types of autofocus systems in modern cameras.
One is the contrast-detection AF, which is used in most point and shoot cameras and in live-view in most DSLRS. This works by moving the lens back and forth until the setting which gives most contrast between adjacent pixels is found. Obviously, this requires actual contrast in the subject — you can't focus on an all-white wall.
The other type is phase-detect AF, which uses a beam-splitter to tell whether patterns of light and dark are back- or front-focused, and moves then lens accordingly. (This is what you're probably using with your Nikon DSLR.) Phase-detect AF also requires the focus area to have a pattern with high contrast in order to work.
There's more technical information about this here: How does autofocus work?, if you're curious.
In your example, the red points which don't work for focus are on a relatively plain gray area. There's simply not enough contrast for the AF system to do its thing. The green points, where you are able to focus, have clear, high-contrast detail — perfect.
The phase detection autofocus on your D7000 works by calculating the offset between an observed brightness pattern seen by a pair of miniature sensors, consisting of single rows of pixels, that make up each AF point.
In order to be able to find a reliable match and thus measure the offset there needs to be a strong variation in the brightness observed over the length of these strips, i.e. there must be a line feature or some other local detail that crosses the strip.
The areas you are focusing on simply don't have enough detail for the camera to find an unambiguous match. There are a couple of other cases when you can have detail and still not be able to focus.
Although displayed as a little box in the viewfinder these AF strip pairs have either horizontal or vertical orientation (or both in the case of a "cross type" AF point usually found in the centre only). Thus if the AF point is parallel to the edge there wont be any variation along the length of the strip.
Finally the boxes displayed are not always very well aligned with the actual AF sensors, so if you have a small detail surrounded by a plain surface it's possible to miss it and AF to fail even though the box in the viewfinder looks like it's right on it!
The problem is probably due to a lack of contrast in the areas you have indicated. They are just flat grey with no shading, which makes it difficult for the camera's AF system to determine when it has focused image.
The green-circled areas, however, have strong shadows and highlights and so the camera can easily focus on them (though if the contrast is too strong it can also cause problems).
If you check out the autofocus section of your camera's manual, you will probably see a section listing conditions where AF may not work correctly.
I find that the focus assist beam from my 430exII flash allows me to focus on bland surfaces like a wall. Perhaps, if you really want to focus on bland surfaces you can try the flash assist beam from a speedlite. You can disable the actual flash firing but keep the beam.
If it is a must to focus on such plain featureless surfaces ,there are some external light sources which throw a light pattern on the plain surface creating dark and light areas, focus can be achieved with such pattern and lock the focus using AF-L button, switch off the focus assist pattern light, meter for exposure and take the picture.
