This depends entirely on the server's rules. Many public Eaglercraft servers explicitly forbid the use of modified clients. If you use one on a server that prohibits them, you risk being banned from that server.
You can connect to any EaglercraftX 1.8.8 compatible server directly through the client's multiplayer menu. Tuff Client vs. Other Eaglercraft Clients tuff client 188 upd
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Set internal render distance between 4 to 8 chunks to avoid bottlenecking low-end laptop processors. This depends entirely on the server's rules