If this is in a commercial scenario, using a commercial graph drawing library like yFiles should be feasible.
With this library you can create your own flowchart editor:
- you can declare and create your own specific shapes and behaviors
- drag and drop, as well as interactive modification and creation of elements is supported
- you can add logic to prevent the user from making certain connections (like connecting the end node to the start node)
- the editor supports automatic snap-lines so that creating a nice diagram is made really easy.
- you can use automatic layout algorithms to rearrange the diagrams and/or the edge routings to get nice flowcharts where the information flow is nicely depicted (going from top to bottom or left to right, e.g.)
Take a look at this interactive set of demos (requires HTML 5 capable browsers) and especially this one. The latter shows the drag and drop feature, the automatic layout using a style that is great for flow-charts and the snapping and some more advanced features like grouping and partition grids.
Disclaimer: I work for the company that creates that library. There are other libraries (although not as powerful in my humble opinion). I do not represent my employer on SO/SE.