I am affraid that those are the correct fingerings, it's not easy. The pinching thumb must cover just the right area or it will squeak or possibly sound an octave lower. The hardest of those fingerings is by far the 3rd c. if, however you get proficcient enough at breath control you will become able to play the e without pinching the hole (and even the f but that's a different story). I do recommend however that you pinch the hole since the tone quality is significantly higher. I can send you a sound sample if you want it.
Oh and btw, cut your nails as short as possible, that will make it easier.
For practise you could always try Brahm's 4th symphony (the oboe version) on here on 8notes. ^^ just try not to annoy bypassers.
Possible you might be blowing too hard. Experiment with different blowing pressure.
Experiment with how you half-way cover the thumb hole. There is more than one way to do it. Some people do with it the end of their thumb, thus requiring the nail to be short. Some people roll or push the thumb sideways toward the far end of the recorder. I hold my thumb so that by straightening the joint, the end of the thumb tips up and uncovers the hole slightly.
One other possibility; I hope this is not it, but it must be mentioned: the little blue recorder suggests that it could be something like a First Act brand, something that is not made to play well. If you can find a music store that stocks the Yamaha "Rottenburgh" 300 series plastic recorders, try one of those. If the squeaks are still present, then it's you; if the squeaks are gone, then it was your blue recorder.
If you must have something colorful, the Yamaha translucent sopranos do reasonably well. They are a bit wimpy sounding, but acceptable.