Why would my liver enzyme tests in a normal person rise to ALT 300's and AST 200's with the flu?

Complicated. One thing which distinguishes influenza from many other viral infections is its systemic manifestations. It would not be surprising to see a low grade hepatitis in association with this. If the pulmonary involvement were severe enough there might be low grade right sided failure with hepatic congestion which could do this. Or you might have something other than influenza.