I reckon most of us have these breaking points, emma - don't feel bad about it. :) IMO, it's not really (or not only) about the 'phone/technology, is it? It's about the feelings of powerlessness! 
The teenage years are all about breaking away from you and becoming independent. Some teens do this quietly and calmly, but most find some 'battle-grounds' they can fight you on.
It sounds like technology has become your family's battle-ground - but it could have been something else like going to school, studying, eating, curfew times, etc...
For teenagers engaged in these battles, the facts don't really matter; what matters is power and control. Whatever their chosen battle, once a teen engages, they want to fight until they have more power and control over their own lives than they had at the start of the battle.
Parents who aren't savvy about this can get into terrible downward spirals. Most teenagers will fight and fight, and have much more energy and determination and motive to win than you do! Some of them will continue to do this even if your response is to go in heavy-handed and 'crush' them, or even if they put themselves at risk by fighting with you (e.g. by running away). If you are not prepared to 'give ground' - by which I mean give them more power and control - then family life can become very miserable, with open hostility, constant battles, nastiness and even violence. :(
If you are a wise parent who knows (and remembers!) what is going on inside teenagers' heads, then the trick is give some ground, while still maintaining boundaries that keep them safe and you sane.
IME this is easier if you think forward to the adult they will become, rather than back to the small child they have been. (That's hard!) If you're tempted to 'crack down', remember that your ultimate aim is not to be policing your DCs' internet use and putting them to bed when they're 37 adult, but to have absolutely nothing to do with these things by the time they are 17-21ish. 
If you think about it, although your teenagers may be looking for a fight, it doesn't need to be a battle for you, because in fact you have the same aim: you want your teens to grow up and be independent as much as they do! (Though perhaps not as fast!
)
The feeling of panic, IME, comes from the fact that as your teens gain control over their own lives, you lose it. You feel powerless because you actually are becoming a bit more powerless - and that's scary.
I think you probably just have to accept this, and find ways of dealing with the negative feelings and being kind to yourself. It's less scary if you realise that this is how it's meant to be.
But remember: you are aiming not to lose control but to hand it over!
In your particular position, emma, I'd do what pootle, usual and outtolunch suggest, and compromise. But don't wait for another crisis, or that compromise will look like you've lost control, and you want to give it. Instead, I'd suggest that you tell your DCs that you have been thinking, and you have decided they probably are mature enough to have more relaxed rules about technology, and you'll be happy to re-negotiate when they're ready to discuss it sensibly.
You decide what you want/need (e.g. no 'phones during family meals, them able to get up in the morning, whatever)... Decide your 'bottom line' (what you must have) and your 'wish-list'. Be prepared to compromise on the wish-list, but not your bottom line.
Then you ask them what they want and need... (I bet you a million pounds it won't be as outrageous as you think!) Encourage them to distinguish between their 'needs' and their 'wish-list'. Help them understand your reasons, and encourage them to explain theirs.
And this book, How to talk so teens will listen... may help :) The authors identify some really useful and important 'principles', I think, including: 'feelings matter'; 'civility matters'; 'punishment has no place in a caring relationship' and (especially relevant here I think!) 'our differences needn't defeat us'. 
Blimey, that was a bit of a mammoth post!
Hope it helps, at least a bit!