Google lets you log in to multiple accounts at one time. The reason that it's asking for your EGCC login when you open your personal email is because the EGCC account is likely set as the default, primary email account. When it asks you to log in to your EGCC account, it will take you into GMail where you can access your personal account as well.
Once inside the account, you can change the current mailbox that you are viewing by clicking on the circle on the top right of your screen, and either selecting a different account, or selecting 'Add Account' and logging in to your personal account. If, however, you want your personal account to be the default account that appears when you go to GMail, you must instead click 'Sign Out' and then sign in with your personal account FIRST, and then use 'Add Account' to add your EGCC email account.
EGCC email login is set to time out after a certain amount of activity (overnight, e.g.), so every so often it might ask you to log in with your EGCC login and password again. This is normal.