The idea of not saying no/avoiding negative phrases is simply a psychology trick.
When you say "No" or "Stop that" the child has to immediately work out what it is that they are doing wrong - they don't necessarily know. It comes with an assumption that the child knows that their behaviour is wrong and are acting maliciously/being intentionally naughty. One of the principles of gentle parenting is to assume positive intent.
So it's better to be descriptive. "Stop running" or "Do not hit your sister" "Don't touch the video recorder" - the problem of course here is that by the time you've got to the end of that sentence, the child has already done whatever it is you didn't want them to do!
So there is a kind of double trick in turning it round to a positive. First, by eliminating "don't" or "stop" you immediately shorten the phrase. Then you've got that old saying about "Don't think about an elephant" - as soon as you've said what you want them NOT to do, the thought is in their head whether they were about to do that or not! Lastly, when you ask them to stop or abandon an action in progress, the problem is that young children are extremely impulsive and their impulses can be very strong. It's very difficult for them to simply stop what they are doing and do nothing instead. Have you ever seen a three year old doing nothing? So think about it - when you tell them to stop, you're actually expecting them either to do something very difficult (replace a very attractive action with nothing at all, which is the most unattractive thing in the world to the point of being impossible for many children) OR to work out, in a split second, what the appropriate behaviour is for that situation, and then do it, even if it's something they don't want to do. This is practically rocket science. SOME children can do it, in predictable situations, when they do happen to know exactly what you want and they are winding you up on purpose, but in most cases, simply reminding them of what you do want, or providing a far more attractive alternative (this is like dog training 
) works much better to get them to cooperate.
So you can be short:
Walk
Be kind
Leave it alone
Or fun:
Let's be bunnies and hop! On this side of me.
Hey quick, I need two helpers! Sam, I need some paper. Amy, I need a red pen.
Look! What's this over here?
Or make suggestions:
I need you to walk on this side of me, or sit in the buggy.
These are Sam's bricks, these are Amy's bricks. Let's move a little bit so you both have space.
Come and press these buttons. You might break that.
Or even turn it around to them:
Wait! We're in the library. How do we move in the library?
Amy, what's happening? What's the problem?
What goes in the video recorder? Yes videos. Not toast. Where does toast go? On the plate or in the bin.
Depending on your child's personality, age, your current mood, etc. It takes a bit of practice at first but it does become second nature - it's quite surprising. It's also useful to teach a generic freeze/wait/stop which you can then use in emergencies.
As for "no" as an answer to a question, it's a different topic. Some people see this as a totally separate thing to an "instructive no" and some people try to avoid this as well. If you choose to avoid or minimise its use as an answer to a question, then it's usually because you think that you should do more for/with your children or trust children more with their own decisions, so it's supposed to counter that situation of DC pestering you for attention and you go "No, no, no" all the time because you're tired or bored or lazy, or children asking for things that you think will be a hassle (like that craft kit you're dreading) whereas you could actually manage it if you spent more than a second thinking about it, and it's supposed to make you think about what you're saying no to and decide if it's really reasonable or not. I believe this is from a book called Taking Children Seriously. Another facet of it is just avoiding arguments, because toddlers will often melt down when presented with a straight No, and if you can get to the same conclusion (not having what you don't want them to have) without a tantrum, why not? So again there are some psychological tricks you can use which help avoid toddler meltdowns when faced with a no. It's been a while so the only one I can remember is the delayed yes - when you say Yes, we can go to the park tomorrow. Right now we need to eat dinner, have a bath and go to bed. Probably you'd also use statements which avoid just being a direct no, like reasoning/offering an alternative: I can't get you ice cream at the moment, but you can have a banana. Hey if we freeze it for an hour, it will be like banana ice cream! Hmm, I don't think that's a good idea, because it would upset the rabbit. What about feeding it a carrot? Or make a deal - I'll play Sonic with you if you can finish your homework and clean your room before 5. Otherwise, we don't have time. Etc. Yes there is some argument that it "doesn't prepare children for the world" but I do think there's a difference between indulging children and saying yes to everything whatever, and carefully considering their request/wording the rejection in a way which is less likely to set them off. It doesn't really matter overly that other adults won't carefully word their Nos in the same way because the main person they are looking to for acceptance is the parent anyway, and most children will be more wary of kicking off in front of an adult they know less well. I know everyone has encountered a child who melts down at every no, but I don't think that's necessarily indicative that they've been parented like this. More likely they just have a permissive parent who gives them everything that they want, or the parent says no too, and they melt down at them as well!