If a customer wants a red widget and all you have is a blue one, and the customer says "but I want a red one", this is not a buying signal for you, but is a buying signal for the salesperson at another company who can get him a red one.
Not all objections can be overcome. Sometimes they can be (for instance, by providing more information for the prospect or clarifying a misconception), but sometimes an "objection" is an indication that your product, your company, or you, are not right for your prospect ("they sent you packing", as you said).