Which barrel is the best, or are they both accurate? Perhaps I'm
comparing apples to oranges, as the Criterion will be stainless
Match Grade while the Shilen would be stainless Select Match.

I have a Shilen barrel on an Encore (Tac 20) which is more
accurate than a couple of my bolts, but am wondering about how
well Krieger constructs their button rifled barrels. Anyone
have experience with the Criterions??

No one here has any experience with a Criterion??

Criterion only on a factory Weatherby it shot fine. But I would have more faith in the Shilen for getting match grade accuracy.
Based on what has been used by the target shooters, I don't think the criterion is used frequently except for the factory class events where the Shilen is more common especially for short range events.

Both button rifled but the Criterion I don't believe is hand lapped like the Shilen.

The Left Hand Accumark I purchased and gave to my son in 7mm STW grouped the first load i tried in it just under 1/2 inch at 100 yards. Nosler 140 gr Accubond and Rel 22. This rifle has the Criterion SS Barrel. If given the choice for a build it would be Shilen or Hart

Give Benchmark barrels a try, you wont be sorry.

These guys pretty much answered it for me:

http://savageshooters.com/SavageForum/index.php/topic,38462.135.html

Note the third paragraph of the first post on that page.