Answers to each part of your question:
I am calculating the probability of default with Chi-squared test Null
Hypothesis [chisq.test() function].
No, you aren't. That's not what a chi square test does. You can estimated the population proportion with the sample proportion and get confidence intervals around that proportion, but this isn't a chi-square problem.
2.- My dependent variable is "default" that has two values: 1 (applicant
defaulted on loan) and 0 (applicant paid loan).
Chi-square tests do not have a dependent variable. They are tests of association. With a dependent variable, you want regression. Since your dependent variable is dichotomous (default/not) you want logistic regression. You can test whatever independent variables you want (e.g. whether the person is a student or not, which you mention below).
3.- For predicting the probability of student (an informative variable) to be defaulted [chisq.test(table(student,default)] which
value should "default" present for applicant defaulted on loan: 1
(True) or 0 (False)?
It's irrelevant which is the default, one choice or the other will make the output a little easier to read, but they are equivalent.