muscleshark, two breaths are necessary. You take the first one with just enough air so that you're solid enough to brace against the weight and take a step back. Then you release your air and take the real breath with the full weight on your shoulders.
If you only do it with one breath and you take that breath before you're under the bar, you're right that you can fill up with more air. However, this is like filling up a balloon nearly to its maximum, placing a heavy book on top of it, and then pressing down on that book. The balloon compresses first under the weight of the book (in your case the bar), causing the pressure to increase, and then the pressure increases even more as you apply force to the book (in your case as you squat). The same thing happens when you take a breath before you bear the load of the bar. Your chest cavity deforms when you have a heavy weight on your shoulders. This double increase in pressure is not desirable and is the reason why you got dizzy.
So, at least, during training:
- always take a breath with the weight already on your shoulders
- take that breath at exactly the last possible moment before you squat to minimize the amount of time you're holding your breath
If you find you can't get a good breath with the bar on your shoulders, there's a problem with your rack position. Fix it.