EVA is fine in moderation, and as others have said it wears/compresses relatively quickly.
For the most part shoes are just tools. There are different types of shoes for different jobs; and any shoe/tool used improperly is going to be less efficient if not destructive.
Running is an interesting sport in that everyone assumes they already know how to run. "I know how to run, it's instinct, it's like walking but way faster." The average person doesn't seek out instruction on how to run they just go do it, but if they want to learn tennis or golf they hire an instructor. Unfortunately, most people don't instinctively know how to run, and when you factor in bad habits, poor posture, gait altering shoes, and sedentary lifestyles you get a recipe for injury.
What I look for in shoes:
1) Do they fit properly?
Your toes should have plenty of room at the ends and on the sides. If your toes are scrunched at the end or squeezed together you need a larger size, or very likely a shoe with a different toe box shape.
Make sure that there aren't any weird points of pressure on your feet and that none of the overlays (the sturdier material often incorporated into the styling of the upper) rub your foot. (for example: if an overlay meets the midsole at your pinkie toe, it may rub and cause blisters during runs)
Your feet should not significantly overhang the sides of the sole. If they do you need a wider shoe.
2) Do they interfere with the function of your foot?
Your feet flex for a reason, and stiff shoes that restrict the flexion of your foot will then displace that stress on to other muscles and tendons. Your shoe should allow your foot to move naturally. Not only will this help to strengthen your foot but it allows you to react to the surface you're running on better and avoid rolling your ankles (think top heavy SUV).
NOTE: Another responder indicated that there is a higher injury rate among barefoot/minimalist adopters. For one thing, I'm not necessarily promoting barefoot/minimalism though I think it has its place; I am promoting allowing your foot to move instead of more or less hinging at the ankle.
As with any fad, many people enthusiastically dove into barefoot/minimalist running despite the adamant warnings of specialists to take it VERY slow. The fact is, the combination of poor footwear that will literally change the shape of your foot over time, poor posture, long periods of sitting, simply not knowing proper running form, and on top of all that diving in too quickly (not to mention those who neglect strength training and stretching) all contribute to an unsurprising injury rate.
3) Do they have a 6mm drop or less? The 6mm number is pretty arbitrary, but you do want a pretty low if not flat heel on your shoe. The reason is that an elevated heel on your shoe will simply get in the way of a proper running stride. It encourages you to land hard on your heel as the rest of your foot flops uselessly down. Instead of artificially encouraging a forward body lean, people instead reach farther forward with their feet and increases plantarflexion. It's not necessarily bad if your heel hits first as long as it lands under your center of mass (which is achieved with a slight forward lean hinging at the ankles) and it is a soft landing followed quickly by your midfoot.
4) Do they fit their purpose? Greater tread an durability for a trail shoe, etc. Don't trail run in a Nike Free, the flexible upper and tall narrow sole will let your foot slide off the side and roll off the edge of the sole, in turn rolling your ankle.
You will develop footwear preferences over time. I would encourage some minimalist running periodically for short distances to improve your foot strength. However, it is not the only way to run.