No, and considering one of World of Warcraft Cataclysmic problems is population imbalance from realm to realm due to no one logically wanting to be the losing side, it only makes to cut out the middle man and randomize Brink's teams every game.
I mean seriously, who ever picked the fat kid in P.E.? If you could choose factions, eventually all of the "good" players would stay on Team X, the people who didn't know that Team X was for the cool kids would get stuck on Team Y, hence, you have team Y perminently getting steam rolled every game.
... The Side is not Random. The characters you create (16 of them) all have a side you choose also specific clothes etc.. depending on the side. So If you have a really awesome looking security dude with heavily unlocked clothing etcc... wtf do you think will happen if the game forces the char your playing as to switch sides, you cant wear security clothing on the resistance side.
You can change characters between matches (like ShadowCat said)
For their to be some type of random side generating you would have to pick 2 characters, one Resistance and one Security So it changes between each char depending on the side.