Personally I would recommend yEd because I have had good experiences with it. It uses the XML Format GraphML by default so your users will not have to use an export-Function but can use the files produced when saving their work.
But I don't think one can answer your question with a simple "I recommend program X." because nobody knows what you really need. Maybe one program does not give you all features you want to have for your flowcharts. Maybe others are not cheap enough.
I think you will have to find the right one by yourself. To determine which program can do it for you could check several drawing applications. A list of flowcharting programs can be found in this question. If that is not enough you may use google to find more.