Squirrel Mail is a web-based IMAP client. If you've been using Apple mail as a POP client, then you're either removing mail from the server whenever you read mail on your Mac, or you've set it to leave mail on the server so you can see it in Squirrel Mail as well. This setup is not good, and DreamHost will limit your inbox to 2000 or so messages because large inboxes put a huge load on the mail server.
I have my Mac set to be an IMAP client so I always have copies of my mail on the server, but I don't leave everything in the Inbox; I've created other mailboxes to save mail.
You can't just turn a POP account into an IMAP one in Mail.app. You'll have to create a new IMAP account, then copy your POP mail into your IMAP account, then sync to the server. Then all your mail will be back on Squirrel Mail.