The button you refer to is not the shutter release, instead it's used to disconnect the film advance mechanism to enable a used film to be rewound and so is called the film rewind button. Unlike later models, it is just a mechanical release, it does not actually rewind the film.

Usually, it will pop back out again if you operate the film advance lever, but your camera is now around thirty years old so it's highly likely that the internal lubricant films have dried out and gummed up the works.

Repair is not really a DIY option unless you are prepared to risk failure and ruining your camera in return for gaining experience. Professional repairers can carry out a CLA (Clean, Lubricate, Adjust) but the cost is more than your camera is worth in purely financial terms. They will also usually recommend a one-off additional fee for replacing the foam light seals and mirror buffer if it's not been done already as the original foam was unstable and will have either turned to corrosive black goo or gone further and dried out and fallen away.

If your camera is in good condition, then a CLA and seal replacement will give you a camera which is a joy to use and which has a huge range of high quality lenses available for free or very cheaply. It can produce results which still beat most digital cameras, especially if you scan the film after it has been developed. On that basis, although repairs make no economic sense, they are a cheap way to get a superb functional and useful camera. before investing in such repairs though, make sure that the two buttons on top of the camera which alter the shutter speed are working correctly. If they aren't, then the camera needs a new flexi circuit board and it's very difficult to replace and spares are almost non-existent. In that condition, your camera is best sold as a spares donor to keep other ME Supers going. Millions were sold, and although still popular they are easy to find in good condition for free (via FreeCycle, or asking friends, colleagues and family) or very cheaply via online resources and traditional used camera shops.

If you repair it yourself, then you basically need to strip it down, clean out all the old gummed up lubricants and replace them with fresh lube of the correct type. It's more difficult than it sounds with an ME Super as there are also a lot of soldered joints, and so if you wish to gain repair experience it's not a good model to begin with. Online repair manuals are available for free from many sources, just enter "ME Super repair manual" in your search box.