Quoting Haudricourt:

For those who were wondering, Brewers outfielder Brad Nelson did not set any kind of record by going 0 for 21 before being sent outright to Class AAA Nashville on Wednesday. According to the Elias Sports Bureau, there is no record in the book for most hitless at-bats to start a season.

The least they could have done is consulted Baseball-Reference.com. Since 1954, the longest streak of hitless ABs to begin a season is Bob Buhl's 70 in 1962. In fact, Buhl famously failed to record a single hit that entire year, ending with a fat .000 batting average in his 35 games. Buhl was a longtime Milwaukee Brave and actually started the '62 season with team, though he compiled 34 of those hitless games with the Cubs (ha!) after being traded.

A couple players registered long season-opening hitless streaks in the next decade, but since 1972, the record belongs to none other than Jeff Suppan, who inaugurated his 2004 season for the Cardinals with 43 hit-free at-bats over 23 games. In second place is another familiar name, Doug Davis, who put up 42 similarly afflicted ABs in 21 games that same year for the Brewers.

Nelson does indeed own the Brewers(/Pilots) position player record for season-opening hitlessness, coming in ahead of the 17-AB streaks achieved by Jim Gosger in 1969, Tony Muser in 1978, B.J. Surhoff in 1991, and Gerald Williams in 1996.

If we remove the season-opening criterion, we discover that the most at-bats over which a Brewers batter has gone hitless is 49, again by Doug Davis, starting in 2003 and incorporating his 2004 streak. Greg Vaughn owns the non-pitcher record with 34 in 1990; Craig Counsell is a surprising second with 32 in 2004.