Does anyone know the answer to this? If a horse qualifies once but is entered for another qualifier and does not qualify due to breaking, can the horse be entered to race or does the horse have to requalify?

EW is correct. It is up to each state racing commission.Â
Below is the Indiana rule for Qualifying Races. Item #8 addresses your concern.
A complete listing of all their rules can be found at
www.state.in.us/ihrc71 IAC 7-1-28 Qualifying Races
Authority:Â IC 4-31-3-9
Affected:Â IC 4-31
Sec. 28Â Declarations for qualifying races shall be governed by the following:(1)Â Â Â A horse that has not raced previously at the gait chosen must go a qualifying race under the supervision of the judges and acquire at least one (1) charted line within the qualifying standards of the track.
(2)Â Â Â A horses that does not show a charted line within its last six (6) starts must go a qualifying race within the time standards of the track.
(3)Â Â Â A horse not meeting qualifying standards in its last two (2) consecutive starts on a good or fast track must qualify.
(4)Â Â Â Horses racing with or without pacing or trotting hopples for the first time must qualify.
(5)Â Â Â When a horse is used for the sole purpose of qualifying a driver, that horse must go in qualifying time or be placed on the list to re-qualify, and the race must be charted.
(6)   If a horse takes a win record in a qualifying race, the record must be prefaced with a "Q". The record will not be considered official unless the horse is post race tested (urine and blood).
(7)Â Â Â The judges may permit free-for-all or invitational class horses to go a timed workout consistent with the time it will race in competition in place of a qualifying race.
(8 )  If a qualified horse is entered into a qualifier, that qualifier will not be considered if the horse fails to qualify. Only one such failed qualifier will be permitted. (Indiana Horse Racing Commission; 71 IAC 7-1-28; emergency rule filed Feb. 10, 2003.)