boolean type for while loop in bash?

No need to you a helper variable, consider using:

while ! mysqldump .....; do :;done

Try

while [ "$VAR" -eq 0 ]; do

0 and 1 are both considered True because they are not null strings.

You could use /bin/true and /bin/false. Although while ! mysqldump .....; do :;done (Jürgen Hötzel's solution) would be better in this case, there are other cases when using true and false are useful.

#!/bin/bash
VAR=true

while $VAR; do
  if nice -19 mysqldump -uuser -ppassword -h database.hostname.com --skip-opt --all --complete-insert --add-drop-table database_name > ~/file/system/path/filename.sql; then
    VAR=false
  fi
done