Basically, anybody who bets a single green chip at either establishment becomes the object of immediate suspicion.
Floor persons at the Eagle have orders from the management not to be so paranoid, but it just seems to be in their nature - for red chip stakes, which I presume you plan to play, you are unlikely to have any problems on six deck (and they mail some cash coupons too).
On the other hand, the Clearwater has been known to kick out players who have either a.) Bought in too much, or b.) weren't counters, just drunk and playing big. A wasteland.