I am having a internet connection by squid proxy web server. When I want to open any website it asks me username and password. Is there any possibility that without giving those details to get internet?
It sounds like there is an authentication service enabled on the proxy. If that is the case you will need to provide the appropriate credentials. Is this proxy managed by someone else? Check with that person.
This depends on who has control over which part of your connectivity. If you control the squid proxy yourself, you can turn authentication off. If you cannot control that squid proxy, you have to live with the fact. But in that case you can still store username and password inside your browser (in the proxy configuration).