I have a macro which prompts the user to input a variety of values,
which values are put into a range of cells for the purposes of
calculation.

Say the range is A1:A7.

If the user hits the cancel button on the pop-up window, FALSE is
returned; this FALSE generates errors in the formulas that depend upon
these cells. Is there VBA code I can use that essentially says "if
any cell in the range A1:A7 contains the value FALSE, replace the
contents of all those cells with "" ? (i.e., an empty string.)

Are you looking for code something like this?

Dim R As Range
Application.EnableEvents = False
For Each R In Range("A1:A7")
If R.Value = False Then R.Value = ""
Next
Application.EnableEvents = True

Rick