This is a more complex question than just a simple yes/no. There are two different effects going on.
First is eyestrain can be caused by your pupils having to dilate and contract. When it's dark, your pupils dilate (become bigger) to let in more light, allowing you to see better in the dark. When it's bright, your pupils contract to reduce the amount of light coming in.
Reading or viewing a computer screen in a dark room is generally a bad idea because it forces your pupils to repeatedly dilate and contract as you look at and away from the screen. I suppose you can think of this as exercising your eyes, but usually it just leads to eyestrain. The best lighting/screen brightness is when the screen is about as bright as a white sheet of paper in the room's ambient lighting. This allows your pupils to stay at around the same size no matter where you look, reducing eyestrain.
The second effect addresses your nearsightedness. The size of your pupil affects something called depth of field. The smaller the pupil size, the more stuff is in focus. This is easily demonstrated with camera equipment.
http://ecvphoto.weebly.com/uploads/1/5/4/7/15476082/8643193_orig.jpg
In the picture on the left, the lens has a wide aperture - the lens has a large diameter, equivalent to your pupils being dilated. This reduces the depth of field so only the monkey is in focus, while the fence is out of focus. In the picture on the right, the lens has a small aperture - equivalent to your pupils being contracted. This increases the depth of field so both the monkey and fence are in focus.
Nearsightedness is simply a shifting of the focal range of your eye. Whereas a normal eye might be able to focus from 6" to infinity, a nearsighted eye might only be able to focus from 2" to 20 feet. The focus range has been shifted closer. These are absolute focus ranges though (what would be in focus if you had zero depth of field). The range of stuff in focus can be increased by increasing depth of field.
When your pupils are contracted and depth of field increases, even though the eye is focused at the same distance (6"), the normal eye may be able to clearly see things as close as 3". A nearsighted eye may be able to clearly see as close as 1" and as far as 80 feet. (Numbers are made up - the math is more complicated than I'm letting on.)
This is why when you don't have your glasses on, squinting can help you see better further away. The squint reduces the effective aperture of your eye, increasing your depth of field, thus allowing you to see further. You can achieve the same effect by holding your finger and thumb very close together in front of your eye. Or most effectively, just poke a pinhole in a piece of paper and hold that in front of your eye. That's actually good enough it can substitute for glasses if you ever break the lenses.
Ok, so what does this have to do with room brightness? Well, remember how the eye is focusing 6" away, but the increased depth of field due to a contracted pupil increases the range of stuff in focus to 3"? Well, stuff 12" away is also in focus due to the increased depth of field. So if you're viewing a screen 18 inches away in a dark room, your pupils will be dilated and you will have to focus at exactly 18" away to view the screen clearly. OTOH if you're viewing the screen 18" away in a bright room, your pupils will be contracted. The increased depth of field means your eyes can focus at 36" away, but the screen 18" away will still be in focus. Focusing at 36" is easier than focusing at 18", so your eyes will suffer less strain viewing the screen in a brighter room.
As for whether it will make your myopia worse, I don't think scientists know. I've seen arguments both ways, and the scientific literature on it is inconclusive. Still, if the theory is true that constantly focusing on stuff close to you worsens myopia, then you want a bright room so your pupils are dilated and you can view the screen or read a book while your eyes are focused at 36" away, instead of at 18" away.