Does one player best responding to sample from a mixed strategy, and the other player minimizing regret converge to a Nash eq in a zero sum game?
No, it's not true. Consider this game where the row player's actions are A,B,C and column's are D,E (shown are the row player's payoffs):

    D     E
A   1     0
B   0     1
C  0.6   0.6


I think that in any equilibrium, the row player plays only C. 

But now if the column player uses a no-regret algorithm, every sample will either be action D or E, and the row player's best response will always be either A or B. So the row player will never play C in the entire empirical history.