Squeezing shorts is trying to make sure that the shorts (that need to buy back shares) don't find any shares to buy, thus driving up the price (as they'll offer more and more to buy the shares they're obliged to return to those they borrowed them from).
With a gamma squeeze, you also aim to drive up the price, but the mechanism is a different.
As you might know, the payoff of a call option (as a function of the stock price S at expiry) looks something like this:
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With S below the strike K, it's worth nothing, and as S exceeds the strike, the payoff goes up.
The price of a call option before expiry looks basically similar, but smoothed out.
The delta of an option is the first derivative of that graph, or, to put it differently, the change in the price of the option as S goes up. You can convince yourself that when S is very low, the delta of the option is very close to zero, and as S approaches K, the delta increases, and with S much higher than K, delta is basically 1.
(Take an option struck at 50. When S = 10, it's worth basically nothing, and when S = 11, it's still worth basically nothing. So, delta = 0. When S = 200, option is worth about 150, when S = 201, it's worth about 151. So, delta = 1.)
[Gamma is the second derivative. It's basically zero when the stock price is very low or very high, as delta doesn't change. But near K, gamma rises - delta changes!]
Now, if an option trader sells you a call, you're long (you win if the stock goes up), trader is short. Trader doesn't like that risk, so they'll buy the stock to cancel out their price risk. How much stock? Well, depends on delta. If S is small, and delta, say, 0.1, they only need to buy 0.1 stock. Then, for small movements of S (say, a small increase), what they lose on the option, they gain on the stock.
However! If the stock moves substantially, particularly if S approaches K, the delta of the option goes up (gamma is not zero anymore). So, the trader needs to buy more shares to be flat again.
So, here's the strategy: Cheaply buy way-out-of-the-money call options (S << K). Then, manage (somehow), to drive the share price S up towards K. The trader who sold you the options will then buy S, potentially driving up S even further. That's the gamma squeeze.
[Note: The trader that sold you the call charged you a fixed premium at the beginning. Now they must buy the stock when it has gone up, and sell it when it has gone down. So, as the share price goes up and down, they lose money (since they buy expensive and sell cheap). If the share price goes up and down a lot, they'll lose more than they got on the premium. If the share price goes up and down only a bit, say when it moves smoothly in one direction only, or not at all much, then they'll lose less than they got on the premium, so they'll win.
That's why we say that someone that wrote a call is short realised vol: if realised vol is high (higher than the implied one that he charged you for the option), they'll lose, and vv.]
When you wrote an option, you're short vol, short gamma, and long theta. When you bought an option, you're long vol, long gamma, short theta.
The gamma squeeze also works going down, if they had to load shares on the way up they have to unload on the way down, accentuating a down trend.
When you drive the price of the stock up you increase the interests the short have to pay on the shares they borrowed to sell, or maybe increase the capital requirements they need to maintain this position on their books, reducing their upside.
If the interests or the capital requirements for maintaining the position get too high, or the owners of the shares want them back, they might have to exit their position, "no matter the cost" hence the "squeeze" part.
(I'm writing it down to see if I understood it all correctly)
You’re right, the option gamma squeeze works on the way down (past the strike) as well in accelerating the movement.
It is as if the option seller (who is short gamma) follows a momentum strategy - when price goes up, they buy, when price goes down, they sell.
As for the short squeeze - why would the owner of the shares want them back from the shorters? Well, presumably to sell them.
The conversation would go like this: BlackRock to Melvin: give me my shares back! Melvin: why? B: the Price has gone up to 300! It’s insane! We are a traditional fundamental investor and bought them at at 15. We want to sell them and lock in our profit! M: ok, how about I buy the whole bunch of you, and we call it even? B: done.
All the redditors that hold on to their shares for dear life: what just happened?
Therefore the gamma squeeze has a timing component, while short squeeze can happen at any time irrespective of when options expire.