When a meltdown hits, the first thing I do is breathe. I've survived the last ones, I'll survive this one. I acknowledge that I'm having a meltdown and that I need to deal with it before anything else.
Then, the hard part is trying to seperate yourself from your emotions. A meltdown isn't quite you. It's your body telling yourself something is wrong and that you need to do something about it. So take a breather (you're not going to get anything else done anyway) and let your bodily functions happen. If you need to cry, go ahead, if you can get some fresh air that often pretty good.
To finish calming down I try to bring some humor and lightheartedness to my emotions, althought its not easy to do.
On the medium term what helped me the most was getting a healthier life style. I tried meditating a bit but it asked a lot of effort to take the time to do it so regularly. On the other hand, exercise helped me a lot.
On the long term, even if you seem to be against it, the best solution seems to be therapy. I understand that you may be against it, however finding a good therapist can really help.
If you all the answers here haven't convinced you to go the psychotherapy way, I'd suggest finding the next best thing. Talk about your issues to some friends or people you trust. This is not easy either. It requires being really open and transparent about extremely complex issues that you keep bottled up even to yourself.
What helped me most was not judging myself for being emotional. Which is absolutely easier said then done. However small issues could spiral out of control because I didn't handle them well, and then I'd hate myself more for not handling them well, and then I'd hate myself for being so negative about myself and being just a liability to myself and the people around me which was just a vicious circle which ended up in huge meltdowns.
For context, a bunch of this advice I got from another answer on HN and really worked wonders. Not everything is better but I'm in a much better place than several years ago. As I said, mental issues need a lot of time to solve, and accepting that is one of the hard things to do.