Do you have any opposition to having them clock in and out manually? The problem I see you running into with trying to automate that is with the 'logging off' piece. In that instance - is closing the page and having a long enough inactivity time count as a firm sign out? Closing the browser doesn't really mean they signed out of Quick Base - so it would be really difficult to try and automate that. Plus automating activity around time can be a little messy in general.
A clock in/out idea might be to have a 'Daily' record for each person - and put it nice and big on their dashboard as a report of 'today' - and the only things on it are two buttons. Sign in and out. So every time they hit the home page they see it. If they forget they can go back and update yesterdays as well potentially.