I think it's possible to prove, under the right conditions. One assumption is, that Humans are not point like, so that depending on the exact position the person may be half on two sides. For simplicity, I assume a continuos finite population density everywhere, which can be a small peak where a person is and zero everywhere else.
In this case, it is obvious that for any angle, we can draw a line splitting the population in half. Imagine just shifting the line until the population on both sides is the same. This means we can, for any angle, draw a cross with each line splitting the population in half. This can be written as the following condition, considering the colors above and R, G, Y, B as populations in the Red, Green, Yellow, and Blue quarters, respectively: R+G=B+Y and R+B=Y+G. So G=B and R=Y. What we still need to prove is that it is always possible to have G=R. Now we can do this continuously for each angle of the cross, so starting with an arbitrary cross, we rotate it slowly 1/4 turn counterclockwise. Now R is where G was before. Due to the conditions it can be exactly the same cross but with colors switched. So if R>G before, we now have R<B=G, so during this path, and everything being continuous, there must be an angle for which R=G holds and so all four quarters are at equal size.
The real question is, does this hold on a globe with great circles is splitting lines?
To add what the others said, this image is most likely taken with a special filter for taking only one specific wavelength, so color. In this case H-alpha, so excited hydrogen atoms, which is deep red. With this and additional filters for safety you can see more or less this image yourself, except it's red. I already had the opportunity to try this.
Here is a site showing daily images of the sun taken with different filters. Red is H-alpha, also shown in OP. Only with this filter you can see the protuberances. White is white, so what you would see if you could look directly without burning your eyes, or what you see with eclipse goggles. Right is another special Line, Calcium K. All of this you can look at with the right filters and a telescope and it looks similar to the images here, except the two colors are even more saturated than shown here. However, changes are on the order of minutes, so it looks more like an still image.
However, the sun and planets are pretty much the only object where images are similar to what you could see with telescope and filters. Colorful images of the moon are always heavily processed. For nebulas and galaxies its even more of a difference, they are just too dark to see more than a grey blob. For this a telescope does not help much, similar to a lens not helping to see in the dark. So nebulas and galaxies are shown at least hat they would look like, if they were brighter. But most of the time they are shown with a lot brighter colors than reality.