Can President Bush run again during this upcoming election?

Nope. He has served two full terms as president, so he is now constitutionally ineligible to run for president or vice-president. The Twenty-second Amendment, ratified in 1951, prohibits a third term and sets ten years as the maximum anyone can serve. An acting president can serve the remaining half of a predecessor's term and then two full terms, adding up to ten years. If the acting president takes over before the half way mark of his predecessor's term, he is limited to one full term. No president has ever served more than two full terms (except for Franklin D. Roosevelt, who was elected to a fourth term in 1944 before the amendment was passed), although Lyndon Johnson could have done so if he had not chosen to step down and defeated Richard Nixon during the 1968 election. Johnson was John F. Kennedy's vice-president, and he took the oath of office in JFK's third year on November 22, 1963, the day JFK was assassinated.

Here is the text of the Twenty-second Amendment that applies: