The problem with the cold method is that it won't extract many of the flavour compounds in tea that require higher heat. Extraction of flavour takes place at varying temperatures and the problem is very high temps will extract bitter compounds, which is why you probably prefer the cold method. I'd much rather use hot water, but far off the boil, for extracting tea flavour. And yes, never squeeze the tea bags.
The major problem with tea bags is that they are made with the lowest quality teas, which is why it is so easy to extract awful flavours from them, and they are always a hit-and-miss - even from the same supplier. Tea grades are determined by colour and size and unfortunately tea bags utilise the lowest possible tea grades as the bags themselves cannot extract flavour from larger tea grades. They use grades call tea dust and tea fanning. As you can imagine, the dust is simply a byproduct of actual tea making and this is what they use in those bags. It really is awful quality, regardless of how they try to market it to you.
If you want great tea, then try to buy the stuff from an actual tea supplier, even if it's just to test it out. Try to get Darjeeling tea, with a grading of at least BOP (broken orange pekoe) or OP (orange pekoe, which is better quality). And never brew tea with boiling water...