As start, are you sure you want to do this really. Using shortcut for current date (for me it is Ctrl+Shift+Semicolon, but it depends on your local settings) in cell in column C is so easy, there is no real reason to take much insecure approach.

Anyway, when you are sure about this, then formula in not a way for this. There is no way to prevent it to be recalculated at some appropriate moment. You even don't need to change anything - it will be enough the user double-click on cell (the editing of cell is activated), and then move to another one, and the formula is recalculated (and the date is replaced with probably new current date).

The only possible way I can think of, is to write a worksheet's Change event, which checks for values in columns B and C of active row, and when B<>"" and C="", then enters current date into cell of column C of currently active row. To make this even more foolproof, the script can check for active cell, and insert the date only when active cell is in column B or C. But the event fires whenever you edit any cell of worksheet, and at least checks for values of cells in columns B and C of active row - i.e. it spends some time for it and your workbook slows down as result.