The webbrowser shows a url like localhost:number or localhost/squish_extension_port_number=number when trying to record or replay a test and stays there for a bit.
When Squish automates a Webpage in a browser this page is used as a method to transfer information from the process that starts the browser (webhook) to the extension inside the browser.
The url will automatically change to the one specified for recording or replay, once the extension has identfied the browser tab with this special url and read the port. After that the communication between extension and Squish happens via network sockets.
If you enter this url in a regular brwoser on the system, this should show an error page or at least keeep the url intact.
Should it not be possible to fulfill those conditions, you can still attach to a running browser instance as shown here