The error 10060 is a winsock timeout error which originates from the client's computer, not from the website. It is a sign that anywhere between their computer and the server the communication is blocked.
On which type of communication does this error occur? Is it while accessing normal site pages, accessing encrypted pages (https://...) or when accessing the site with another protocol, for example FTP? If the ISA server in between has been configured to allow only certain types of outbound traffic, it may be blocking the type of traffic you need. If the IT department has run a traceroute, they should be able to tell you at which point their test stops, i.e. somewhere near their own network or close to your hosting provider.