[SalesForce] Can you add a custom button on delivered Report Page in Salesforce.com or custom link

No, you can't do this with a custom button. Custom buttons are not intended to be used as links and probably won't work at all. The links from the custom buttons are the result of a post and just copying the link will lose other information that is passed a long with the button click.

Depending on your use case it sounds like you'll need a visualforce page. You're external system would send users a link to the visualforce page with whatever parameters you need to process on the salesforce side. Unfortunately, your use case sounds a little unusual and it may be better to ask a separate question about your specific use case rather than the more general can I pass in parameter question.

Note however, that if you want to pass in attributes from a salesforce record to the url resulting from a custom button this is easily done by editing the content source of a custom button.

Step 1: add a custom button to Contact object by Setup -> Contact -> Buttons, Links, and Actions -> New

Step 2: go to contacts tab and click edit layout link and add your button to page layout.