Попробуйте это.
function restUpdateByemail(Request $request,$email){
$students=student::where('email',$email)->first();
$students->name =$request->input("name");
$students->password=$request->input("password");
$students->gender=$request->input("gender");
$students->country=$request->input("country");
$students->save();
return response()->json($students);
}