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Does MSE loss function work in NN training for predicting values between 0-1?

To summarize all the comments and what my original mistake was: The sensitivity effect is still there even though yes the squared of the error value less than 0 is less that itself however: ...
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Does MSE loss function work in NN training for predicting values between 0-1?

Depending on what you want to do, there are advantages to other loss functions (crossentropy) and other regression models (beta regression), but there is not necessarily a reason to dislike MSE as a ...
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Why do the training and validation loss curves diverge?

You have to keep the training loss value constant. Beyond that, your teaching technique or network model is not suitable for the job. I recommend that you are using MLP.
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