Most barrel races or rodeos do not have any rules with the type of tack you must have. (And if you are unsure, you can always ask the association you want to run with.)
I myself prefer a browband headstall with a throatlatch. I feel it is more secure; not only for barrel racing but for any time I ride my horses.
A breastcollar is not a necessity (if you saddle fits well like it should) but it is a nice piece of extra equipment to have "just in case" something else breaks. There has been occasions where a cinch strap will break, and the breastcollar helps hold the saddle in place until you can safely dismount. But there are some horses where a breastcollar might not fit their body well and can actually hinder movement.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.