Besides everything else that's been mentioned, if you missed a PDX-EUG flight there's no way you could drive to EUG in time for the same connecting flight (the drive is about 2 hours, plus any heavier-than-normal traffic time on I-405). Even if you tried to claim this, and UA was willing to bend a rule, the agent would know you were making it up.

Of course, this is the same staff that put my son's EUG-PDX-ORD suitcase on an Allegiant Air flight to LAS...