How do you want to drive your car is the question.
Your differential and axles are not all that strong having been designed to handle the power of the old stove bolt 216 cubic inch six cylinder. So throwing 300 to 500 horsepower at a rear end designed to handle only 85 horsepower will result in carnage.
That being said: yes! Yes boxing in your control arms will stiffen them up removing flex. Of course if you retain the old worn out rubber suspension it may not help all that much. You can replace the rubber bushings with polyurethane ones to provide more control.
A new adjustable Panhard bar is almost required if you raise or lower your car. this is because your rear end swings in two different arcs at ninety degrees to each other. The most obvious is the arc the rear end swings through as the suspension goes up and down. this moves the rear end forward and backwards in the car (approaching the front wheel arch in the outer fender as it does so). Additionally as the rear end is attached and held in the center of the car by a Panhard bar the rear end moves from one side of the car to the other as it goes up and down.
Following this arc can move your rear end into the outer fender lip on one side as it draws away on the other. If the Panhard bar is adjustable it can be shortened or lengthened to center your rear end under the car from side to side.
As to buying aftermarket trailing arms it's strongest advantage over a stock trailing arm is the ability to adjust the pinion angle. They are stronger than even boxing the stock suspension, but only because the control arms are made out of a thicker walled rectangular steel tube. If they use a steel heim joint (aka Johhny Joint) it can transmit a lot more road vibration into the car (solid steel versus plastic or even softer rubber that would absorb some of the vibration).
I replaced my stock arms on my Impala only after ripping one of them off the car body under hard acceleration (stock parts were never designed to handle the abuse I was putting them through since I had increased the power of the motor by 650%).
I recommend boxing in your factory control arms, replacing the rubber bushings with plastic and adding an adjustable Panhard bar. I also recommend adding a rear sway bar but most importantly it is your shock absorbers that will make the greatest difference in your cars handling. I know you bought new "Performance" shocks but what you need are double adjustable shocks to tune the car's suspension (or at least a five click single adjustable shock).
From drag racing to road racing, to circle track or Autocross, being able to tune the shocks to the track is what separates the winners from the losers. If you watch any drag race on TV you will hear the commentator state that the loser didn't have the right set-up (suspension adjustment, or clutch adjustment). He rarely states that he didn't have enough horsepower (which separates only those top end finishes where the competitors are neck I neck).