Would it be correct to say : ‘Just stay alive and wait patiently until the government officials [color=red]come to get you’ instead of ‘come and get you’?
Yes. In cases such as this, where the sense is “in order to”, “come to ~” is preferable in formal English, in my opinion. “come and ~” is widepread in everyday English.
Note that “come to” is usually not natural in the imperative. For example, you would say to someone “Come and take a look”, not “Come to take a look”.