What type of email account do you have? Webmail or POP with an email client?
If it's webmail, such as gmail, Yahoo Mail, or any email account you access via your web browser, your question doesn't make sense to me because your email resides on a email server somewhere on the Internet and can be accessed by you anywhere that you have Internet access. I guess this is the type of account you have since you mentioned your web browser, but please confirm that.
To put your email messages on a USB drive, this would normally entail having a POP email account and downloading your email to your hard drive. In this situation, you would have options for saving messages on a USB drive.