What is the difference between type I error and type II error in forecasting default of a company? For example, I can set null hypothesis to be "The company will default next year", or I can set it to be "The company will not default next year", so my type I and type II error will be vice versa in this case. Can I set the null hypothesis as I want, or how should I decide what is the null hypothesis?
What is the difference between type I error and type II error in default or non-default case?
In general, you want to reject the null hypothesis so you set it as the case you want to avoid. When reading $p$-values and probabilities, it's easier to interpret the levels you want to define as showing stronger evidence against the null hypothesis, but it is up to what you consider the outcome you want to avoid or show.
In your example, I'd write:
$H_0$: "The company will default next year"
$H_1$: "The company will not default next year"
Type I error : Forecasting $H_1$ when $H_0$ is true, also called a false positive
Type II error : Forecasting $H_0$ (or more precisely failing to reject $H_0$) when $H_1$ is true, also called a false negative
You may see here for another example.