I find solution that instead of using ListBox I used ListView.It allows to change list items BackColor.
private void listView1_Refresh()
{
for (int i = 0; i < listView1.Items.Count; i++)
{
listView1.Items[i].BackColor = Color.Red;
for (int j = 0; j < existingStudents.Count; j++)
{
if (listView1.Items[i].ToString().Contains(existingStudents[j]))
{
listView1.Items[i].BackColor = Color.Green;
}
}
}
}