If you can import the EML files into a mail client that support IMAP this will be easy.
It seems that Thunderbird can do this, but I'm not 100% sure. (Don't have a local copy to play with.)
Anyway, set-up Thunderbird to be an IMAP client for Gmail. (Lifehacker has an excellent guide to do this.)
Then, import your EML files into Thunderbird.
If necessary, move them to a temporary folder (called "Zimbra" perhaps) which will be a label in Gmail. Then just wait for IMAP to sync them all to the cloud.
After that you can delete Thunderbird, keep it around to make local archives of your e-mail, or keep using it as your main client.
I was able to do this a couple of years ago with thousands of messages from my Pegasus Mail client. (There was some pain in converting to a format that Thunderbird could read, but getting the messages into Gmail was the easy part.)
The key is to labels and archive.
Every new message starts with the inbox label and the all mail label.
New labels can be added via filter when the email arrives, or manually by selecting a message and either adding a label via the label drop down, or doing a combination of adding a label and archiving by using the "Move to" drop down.
A powerful part of gmail is the fact that each message can have multiple labels. This is better than a folder.
If you archive a message/conversation it strips the inbox label from the message leaving the other labels intact. When you start archiving you are
To move them from the front page of gmail you must archive them. The labels on the side of the screen are a just an easy way to see all messages that have that label. It will find messages that have the inbox label, those that only have that label, and ones that have multiple labels.
The move to inbox command you see actually adds the inbox label to the messages, which essentially un-archives the message.
Best Answer
Click the "gear" icon on the right of Gmail screen and choose "Configure Inbox". Uncheck "Updates" and any other tabs / categories you do not need.
With only "Primary" tab left, inbox returns to its old behavior: all new messages appear together in the inbox.