While on a business trip using my personal vehicle, I blew a tire. I needed a tow to the repair shop. Can I claim the tow bill as a business expense?

If you are using the actual expense method, include the repair and towing expenses in your calculation of actual costs.

If you are using the standard mileage method, it already includes an allowance for repairs so you can't include them separately.  (In most cases, the standard mileage method is the bigger tax break even if you have repairs.)