I designed an experiment using the fractional factorial design feature in Minitab. Can the results of the experiment be analyzed using something like linear regression and backward selection or is it best to use software that is aware that the experiment is a fractional factorial design experiment?
All the 2 levels designs can be computed just by pen and paper with the need of no dedicated software. Of course a more suitable approach is to use at least a spreadsheet. You simply need to multiply each variable's column by the column of your response. Then the average on every column will be the effect of your factor. If you want to do it faster, but always for free, you can use R with the package DoE.wrapper or RcmdrPlugin.DoE.
The calculation I've suggested you is, for 2 levels design, equivalent to linear regression analysis and variable selection has to be performed comparing the effects on each parameter with your experimental variability multiplied by a proper $t$-value.