The Amber (yellow) LED is the Ethernet link light and is directly connected to the Ethernet PHY. Under normal operating conditions there is no way that the Connect ME can effect the state of this LED. As for the Ethernet activity LED, it is used for diagnostics messaging during bootup and subsequently control is passed to the PHY for use as activity.
Is this occuring on our provided development board or a board of your design? If this is on your board can we test an ME module off of Digi's board?
Are you seeing this flashing behavior when the ME is connected to a hub/switch?
Is it connected to a hub or switch?
If it is a switch, is the switch configured for auto detection? Can you reconfigure the ME's port on that switch to manual and see what happens.
Has Erik has pointed out the error codes outputted on the activity LED are displayed in a three count pattern, i.e. 4-1-1, 9-1-1, etc. These patterns mean something and I can provide these codes meaning.
The fact that the link LED is also blinking would seem indicate that the problem is likely with the module or (if so mounted) with the board you designed.
I understand that you are already working with tech support, so I shall cut my comments off here.