The Weinsteins, who have denied rumors they’re having cash-flow problems, will rush Michael Moore’s “Sicko” into theaters on June 29, just six weeks after its premiere at the Cannes Film Festival. This expose of the American heath-care system — which The Post exclusively reported includes footage of ailing 9/11 responders the filmmaker flew to Cuba for treatment — will be going up against such July 4th weekend offerings as the much-hyped “Die Hard: Live Free or Die” with Bruce Willis and the latest Pixar blockbuster, “Ratatouille.” Somehow we don’t think the Weinsteins and their distributor on “Sicko,” Lionsgate, have managed to set up a tie-in with McDonalds on this one.