To summarize all this, we can see that the type of product you choose plays an important role in answering that question – “Can brainwave entrainment damage my brain?” My personal advice as a long-term user of the technology is to Be selective about what brainwave entrainment audios you download from the internet, and make sure to follow the instructions! 
Because the brain filters and interprets reality in a split-brained way, we tend to see things as separate and opposed, rather than as connected and part of the oneness spoken of by the great spiritual teachers (and, in the last few decades, by quantum mechanical physicists). Thus, at a deep level, the dual structure of our brain, in conjunction with brain lateralization, predisposes us to see and experience ourselves as separate from, and often in opposition to, the rest of the world—instead of experiencing the elegant interconnectedness between us and everything else. Our childhood associations and programming build on this inborn tendency by training us to seek this and avoid that, to move toward pleasure and away from pain, to do good and not bad, and so on. The greater the lateralization in the brain, the greater the feelings of separation—and the greater the feelings of separation, the greater the fear, stress, anxiety, and isolation.

And this is exactly true. Have you heard about walnuts improving your memory and being extremely healthy for you when eaten raw? A doctor can confirm this information for you. Try eating too many of them, especially in the afternoon – you will end up nauseous, and if you have a relatively low blood pressure, you can even drive yourself to a nearly unconscious state. Except for their healthy benefits, walnuts tend to lower blood pressure, and when eaten in excess, they can have some very nasty effects on your body. Does this mean you should never eat walnuts, and never benefit from their wonderful qualities? Of course not!
In physics, entrainment is the process of two oscillating systems coming to assume the same periodic rhythm, such as is observed when two clocks slowly synchronize their ticking and tick together in harmony after some time. Pendulums also achieve this same synchronicity when swinging in close proximity to one another, a phenomenon first observed and written about in 1665 by Christiaan Huygens, a Dutch scientist.
I wouldn’t personally recommend listening to delta wave frequencies for depression, so I’m not sure who advised you to do that? People with depression usually have a higher ratio of theta and delta wave activity, so I would normally recommend listening to high alpha wave and low beta wave frequencies, to help balance things. I have some 10Hz alpha tracks for serotonin release, which you can try for free on my YouTube channel here: I also recommend you give these SMR (low beta wave) tracks a try: I hope that helps.
Ultimately the only way to know if this is right for you, or if it works for you is to test it. A good audio creator should be willing to let you give their audios a test drive to see if it’s right for you. When you test this type of technology, one that is largely left to subjective experience, I would highly recommend you listen to the same audio daily for a full week or two, taking one day off per week. There are many different experiences that can be had with this type of technology, including not noticing it doing anything at all. Having at least 5-10 experiences with it will give you a reliable way to gage whether or not this is going to work for you.
The Brain Wave app uses advanced sequences of binaural tones that are combined with configurable ambient nature sounds and atmospheric music (or a person’s iTunes Music) to stimulate specific brainwave patterns.  It includes powerful binaural sequences for sleep, focus, energy, positive mood, confidence, anxiety, stress relief, relaxation, yoga, memory, motivation, headache, concentration and meditation and more.
After one month of using Equisync, I feel relaxed, calm and not stressed at all, despite that I am leaving my present job soon for an early retirement, I handle the situation very well. I have the feeling that I’m going to be ok and there is nothing to worry about. This is what I have gained so far from using Equisync and it’s really meaningful in the erratic world that we live in.
The fastest way to make brainwave entrainment technology work for you on your path of spiritual development is to pull back ever so slightly once you realize you have reached the peak of your nervous system’s capacity to recover from the stimulus. Because your nervous system is like a sine wave, when it is about to go into its valley, you pull back just a little bit, thereby giving your nervous system, energetic system, and hormonal system enough room to pull out all of their recovery powers. This allows you to recover from stimulus more quickly, and then you get past the waning stage and back to the waxing stage, where you can start pouring on the juice a little bit more again.
It may be that you had the volume too loud, but I would expect you to hear the effects of that straight after you’ve stopped listening, not on a day you haven’t used them. It might be something similar to muscle memory, where you suddenly remembered the sound and sensations it gives you as if you were hearing it again. I don’t know how long you’ve been using this type of thing for, but maybe it’s something that will settle down and disappear once you become more accustomed to the sound.

Subsequently, the term 'entrainment' has been used to describe a shared tendency of many physical and biological systems to synchronize their periodicity and rhythm through interaction. This tendency has been identified as specifically pertinent to the study of sound and music generally, and acoustic rhythms specifically. The most ubiquitous and familiar examples of neuromotor entrainment to acoustic stimuli is observable in spontaneous foot or finger tapping to the rhythmic beat of a song.