CJ- you are correct about the media itself being the better option. However, squeezing will still gives one a better start than doing nothing. Sometimes it isn't possible to move over the media, and squeezing is the only option. I have used this method before.
The single most important way to speed any cycle is to add bacteria at the outset. The more one can add, the faster the cycle will be in place. This is the case for two reasons, the most important of which is to have some number of nitrite oxidizing bacteria present at the start. The other reason is the simple math of doubling.
10x2 = 20 5x2 = 10
20x2 = 40 10x2 = 20
40x2 = 80 20x2 = 40
80x2 = 160 40x2 = 80
The difference between the two totals gets really big really fast. (Bacteria numbers are in the millions.)
If you have nitrite oxidizers in any number at the start, this really jump starts the second stage of the cycle. And here, where the reproduction rate is slower, you are really getting a boost. With decent seeding, nitrite will not spike the way ammonia did. As the AOB start to make the nitrite, the NOB are there to start consuming it before it can build up. Ammonia only gets high because we put a bunch in all at once.
Jess- there is no way to know how much bacteria you got into the new tank. What you can know is that your cycle should be faster than without it. Get the test kits and an ammonia source and take the steps to get your tank cycled. Patience is the hardest fish keeping skill to master. Cycling isn't a race, it is a way to keep your fish healthy by creating a good environment in which they can live.