How long are Brazilian citizens required to serve in the Brazilian
army if they are drafted?

Hello.

One year, generally.

sources:
"Conscription: Twelve months officially; usually nine to ten months in
practice, but can be reduced or extended by six months."
source: Brazil National Security Summary
http://www.exploitz.com/Brazil-National-Security-cg.php

"Brazil
Conscription
...Those who serve generally spend one year of regular enlistment at
an army garrison near their home. Some are allowed six-month service
terms but are expected to complete high school at the same time."
source: Country Data Brazil
http://www.country-data.com/cgi-bin/query/r-1804.html

"...Serviço Militar Inicial ... em caráter obrigatório..."
("...Initial Military Service of obligatory character...")
"...O EAS terá duração de 12 (doze) meses, em duas fases..."
("...EAS has a duration of 12 (twelve) months, in two phases..."_
source: Exército Brasileiro
http://www.exercito.gov.br/02Ingres/menssupt.htm

----
search strategy:
brazil, military, conscription
Exército, "doze meses", site:br

I hope this helps.

Great answer and additional back-up information!
Thanks!