First, I am a reasonable experienced dancer. I prefer DBD plus, but enjoy regular plus and mainstream. I am quite capable of blowing a call (any call) but generally I think I give help more often than I need it. I am of the 'never stop' school. Same of the most fun I have had has been in squares where we know we were 'out' but kept laughing and dancing.
I would insert one general rule for any and all recovery situations: for heaven's sake, KEEP SMILING!
In the broad course of human history a busted square really is pretty trivial. If a square is breaking down, having the dancers who did not make the mistakes visibly upset only makes it worse. On the specific techniques: (please understand I am only presenting my personal thoughts here and I cannot speak about 'A' or 'C' dances. Also, I am thinking about the situation at dances, not in classes)
1. Square your set. I do not think this should be taught or used. Ever. It is not necessary since 'make lines' is just as easy and gets us dancing again sooner.
2. Make lines. Sometimes this is necessary when too many of the people in the square have gotten confused. I think it should be taught (or refreshed) at every level. It is amazing how many times I have been dancing with relatively experienced people, who could not make two parallel lines. This is a useful technique, because most callers are very good about getting the other squares into lines as soon as possible after one square has broken down. Frequently the caller will say something like "we will be with you in a minute", so the broken square knows they've been noticed.
3. Self fix. I do this frequently either by seeing a hole in the formation or by having my brain (belatedly) catch up with my feet and tell me where I should have been. It does assume a certain comfort level with the calls und formations; usually there isn't time to consciously think "well I started in such-and-such a place and the call was thus-and-so, so I should have ended up in a certain place". However, it doesn't take a lot of thought to know that three people in a wave on one side of the square and five people trying to make a wave on the other needs to be 'fixed' (well, usually). When I see something like that, I don't worry about whether I made the mistake; I will fill the hole if no one else is headed toward it, especially it is obvious that the person, who made the mistake, isn't going to move and helping that person isn't possible. (See 'never stop')
I think that trying to figure out where you should be can be discussed (if not 'taught') beginning at the mainstream level, and certainly should be reinforced during workshops at all levels. If the dancers are having trouble in a learning situation take advantage of the problems to demonstrate ways to 'fix it'. After all, being able to see formations is part of understanding and executing calls. Square dancing, like life, is not always perfect; realistically, fixing things when they get 'broke' is necessary. I know it is something we dancers talk about after a difficult tip and joke about after making a stupid mistake.
4. Help someone. I am always grateful to any other dancer who helps me get into the correct position. As a female I think it is often easier for me to help other people than for a man. Folks are less likely to be offended if I gently tug their hand or even put my arm around their waist to move them in the correct direction (or get them to stop moving at the right time facing in the right direction) than if a man did the same thing. My experience is that people who blow a call appreciate someone helping them. But the 'smile' rule is especially important when correcting someone else's mistake. 'Helping' is much easier if the 'helpee' has an understanding of 'self fix'. I do not agree that it implies a low degree of dance skill on the part of the helpee. It just means someone goofed.
Helping techniques also depend on the other dancer. There are some people we have been dancing together so long we expect a little pushing and pulling when necessary. (I mean that we sometimes exert some physical pressure on each other, but not enough to cause an injury.) But that is obviously a special situation. It is often difficult to decide on the fly how to move another dancer into the correct position, or whether to even try. (Sometimes the square is so badly down, or the individual is so confused, that trying to 'fix it' is a waste of time.) There are many other situations in dancing, which are potentially uncomfortable when the dancers do not know each other or the local 'norms' (e.g., swinging do-sa-do; flourishes on weave the ring; slapping hands on star thru, etc). Of course, non-physical help, such as pointing in the direction a person should go or calling to them (cheerfully) can be helpful, if the person who is making the mistake happens to 1) be looking at and/or listening to the 'helper' at that moment, and 2) know they are making a mistake.
5. Follow opposite. Gee, I wish someone had pointed this out to me years ago. Or maybe I should have figured it out on my own. I will start paying attention to my opposite from now on, and I expect my recovery skills to improve. Why wait till advanced to show people how this works??
6. Counterparting. I often will look at other squares to see what formation my square should be in, if more than one or two people in my square are having trouble. If we are doing "mill around" (a call which is never taught but most dancers learn anyhow) and the other squares are in right handed diamonds, it is sometimes possible to get the square dancing again without having to resort to 'make lines'. However, I have never been smart enough to keep track of what my counterpart in another square is doing, assuming I were confident that person was more likely to be in the right place than I am. I don't see any reason to teach counterparting.
7. Never stop. This is my basic philosophy. I don't see what difference it makes, if I am personally in the 'incorrect' position, as long as the formations are correct and we can keep dancing. We may be 'out of position' in relation to our opposites or counterparts in other squares, but we are 'in position' to be able to do the call. This is, obviously, a lower level recovery skill than being able to get all eight people into the correct position in relation to each other, but I am not greedy. (To be honest, I may not even know I am not in the 'correct' position, at least for a while) I don't have a problem undergoing sex changes during a tip, but if more than two people are switched it can be difficult to know who does a 'boys run' (looking at other squares helps with this). I would much rather do a reshuffle on promenade home than stop dancing because I got switched with another dancer. This is less elegant than being in the right place in relation to my opposite or counterpart, but it is often the best we can do, when the square is in trouble. I also think dancers should start learning these skills as soon as they start doing DBD. Why is it only desirable to keep dancing in challenge tape groups??
I am a little defensive about the suggestion that at mainstream and plus we should learn the calls well and not make mistakes. The best dancers I know make mistakes; I've even seen callers make mistakes. The point is that we have fun and we help each other and we fix it and we keep dancing.