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For your purposes, yes, because a loop is more simple and easier to understand (as well as a handful of other reasons around complexity).
In my experience, recursion is most powerful when you are...

A potential change to your current code (so numberEntered stays within the scope of the method):

public static int studentNumberEntered(Scanner sc) {
int numberEntered = 0;
try { ...

You are getting a stack overflow because of infinite recursion (a method calling itself forever with no exit-case). This would be occurring if your catch block is always being hit. My guess would be...

You could do what Norm said, but a potentiality "better" solution (because it does not require the creation of exceptions) would be to do something like this:

if(sc.hasNextInt()) {
int...

If you were using a GUI instead of the command prompt, there would be a handful of ways to do this.

As for preventing the program from crashing, use and if statement and the Scanner's hasNextInt()...