Hey,

On a general note, yes, you can become a government teacher in any school with integrated B.SC B.Ed course. There is no as such provision which pars integrated course students to not become a government teacher. However, you need to note that, mere having a degree do not make you eligible to be a government teacher. There are many central and state level entrance exams conducted for recruitment of government teachers. You need to appear for them and meet the desired cutoff to be a one.