Selected Answer
Short of using VBA there is no way you can change what a user has entered in a cell, even if it is done by pasting. There might be a way to do what you want using Paste Values, but that won't be less complicated than what I propose below.
Presuming that your numbers are in column A, I suggest to add a blank column B, place a formula in that column, copy down, change formula to values and delete column A. This is the formula in column A.
=ABS(A2)*-1
To convert the formula results to values, select the range with the formulas, press Ctl+C (or click 'Copy'), Click 'Paste' -> 'Paste Values' and end with ESC (escape key) to exit Cut/Copy mode.