Yes, as long as you use reputable sites. Because it runs in a "sandbox" (your browser), the game cannot access your computer's private files. However, always be cautious about clicking external links in server chats.
To connect to a working Eaglercraft server, players will need to enter the server's IP address or hostname into their Eaglercraft client. They can also use the Eaglercraft server browser to find and connect to servers. working eaglercraft servers