While most of the reaction to France's rejection of the EU Constitution has been political, I thought I would add a short analysis to what this means to American businesses: nothing.
The movement to the Euro, which is probably the biggest change to business, is already completed. There were probably going to be another layer of taxes and regulations that were to be imposed as a result of a Constitutional EU, but they are already high in Europe and would have been incremental (and the individual countries will still be piling them on, so no net savings either way, really).
Actually, it is somewhat good news to American companies. The bonds that tie European competitors down - expensive labor costs, difficulties in shedding workers, high regulatory hurdles - will now have no chance of slowing. It's interesting that some polls showed that these very items, which have created double digit unemployment in France, were part of the reason that the French voted against the constitution: the fear that the constitution would increase competition. They seem to fail to note that businesses there are shedding jobs like crazy anyway and if they don't open up to competition the trend is just going to continue.