It's a staph infection that is a real mouthful to pronounce -- methicillin-resistant staphylococcus aureus infection (methicillin is an antibiotic) and people usually get it when they're in the hospital. Sometimes there's an outbreak of patients in hospitals and it can be tough to get under control.
This is why people shouldn't run to the doctor for every little sniffle or cold. When antibiotics are given over and over again, the body gets resistant to them and allows a staph infection to develop more easily.
I agreed with a lot of what you said in terms of advise, but your understanding of the science behind it are a little incorrect...
It isn't the person's body with an infection that become resistant to the antibiotics, but rather it is the Staph bacteria that have become resistant to it. It is like a mini-evolutionary process that is happening to the bacteria... When you give a person penicillin to treat a Staph infection, you are putting stressors on the bacteria's environment. If at the end of treatment there are any survivors, it may be that those bacteria were somehow genetically different (through mutations and / or other genetic changes) that gave them an advantage in surviving the antibiotic. If these surviving bacteria are then allowed to grow, they will pass this advantage to their offspring. The bacterial growth will be diagnosed as an infection, so the doctor prescribes more penicillin, and some more bacteria may survive this treatment perhaps with more genetic changes that give them even more defenses against penicillin. The cycle continues until finally penicillin has no effect on the infection. At that point, the bacteria are considered to be an MRSA strain.
The development of MRSA's may also be aided by the misuse of antibiotics. For example, a person that had a previous infection may get another infection, but instead of going to the doctor to get the correct antibiotics to treat this new illness, the person uses the "left-over" pills from the previous infection with the mistaken assumption that any antibiotic will work with any type of infection. Also the fact of a person having "left-over" antibiotics is a form of misuse... When a doctor prescribes an antibiotic, the person is to take all of the pills prescribed. Even if the infection seems to have disappeared, there may still be a few bacteria that are alive, but are still responding to the treatment. By stopping the treatment prematurely (eg stop taking the antibiotics when the infection disappears), you are allowing those bacteria to survive even though they may have been killed by the full treatment.
MRSA bacteria are resistant to a number of antibiotics that have a similar action of effect on the bacteria, which is why it becomes so difficult to treat; it isn't good enough to switch from penicillin to oxycillin (or some other -cillin antibiotic) as you might do if a person is allergic to penicillin since they all act in the same way to kill the bacteria.
And because the infection is contagious, the evolution process doesn't have to happen in the same person's body; part of the evolution may be that the infection gets passed to a different person that has Staph with other genetic differences that then get passed on to future generations. Because of this fact, the amount of people going to hospitals with Staph infections, the amount of antibiotics used in hospitals, and people with lowered immune systems in hospitals, those hospitals become breeding grounds for MRSA's. So even if you don't go to the hospital for antibiotics, it is still possible to get an MRSA infection.
My DIL takes my grandson to the doctor for every sniffle and he needs stronger and stronger antibiotics to cure him now. I kept telling her not to get him on antibiotics all the time as he'd have problems when he was older but she didn't listen. He gets sick all the time in the winter now. She never let his antibodies "grow" to fend off the bad viruses and I'm afraid he will have an auto-immune disease in the coming years.
Taking a child into a hospital all of the time for any type of injury / illness may induce the child to develop MRSA infection, but you've hit on another concern about over-hospitalization: development of an immature immune system. When a person comes in contact with a foreign entity inside the body (eg a bacteria), the body may elicit an immune response which means that the immune system will try to construct its own antibodies in order to kill the infection. But the immune system that has never encountered such an entity before may take some time to try to figure out which antibodies will actually work. If a person is treated with an external antibiotic before a person's own immune system is able to make their own, then when the person encounters that infection again, it will have to essentially start over trying to find an antibiotic. This is also why parents should not be so germaphobic in trying to keep their children "sterile" by constantly having them wash their hands and / or using antibiotic lotions / hand-gels.
Now this type of immune system disorder is called an immature immune system because of the lack of antibodies used to treat various infections including some fairly common ones. However this is not an auto-immune disorder... An auto-immune disease is when a person is "allergic" to their own body. The person's own immune system elicits an immune response against a non-foreign entity (ie a part of the body that is supposed to be there), and attacks the person's own body as though it is a true infection. Although most auto-immune disorders are an innate trait of the person that they are born with, it is possible that an infection may make itself to look like a regular cell such that when the immune system develops antibodies against the infection, those antibodies also accidentally target regular tissue. So while taking a child to a hospital for every little issue may cause the child to develop an immature immune system, it doesn't lead to the child developing an auto-immune disease.