I don't think you will find that there are any emergency
prachutes with square canopies, they are all round.

Not true. Both Relative Workshop and Strong make emergency rigs that
will accept square reserve canopies, and will sell them to you if you
convince them you know what you are talking about when you ask for one.
They're the only kind I use.

Square parachutes while offering more control
when open require the wearer to be in a stable position
when the chute is deployed, they are therefore less
reliable when deployed in anything other than a stable
attitude.

Not true. Square emergency parachutes are used with freebags (they
won't even have a bridle attach point), which allow deployment even if
the pilot chute or bridle entangles with the body of the jumper due to
an unstable opening. The same is not true of rounds, which are thus
more prone to 'horseshoe' malfunctions. Picture what that looks like -
if you really want to.

The Irvin EB** series were reputed to be the fastest
opening chutes in the world at one time and they certainly
use round canopies (I24). Maybe they still are, I hope
so cos that is what I have.

The primary factor affecting the speed of opening (assuming such
variables as airspeed, altitude, etc. are kept constant) is the volume
of air required to inflate the canopy. Square canopies need less air
to inflate, as they are smaller. The can be made smaller since they
generate lift, not just drag. Because of this, squares inflate much
quicker than rounds designed to carry the same loads - so quickly in
fact that all modern squares (other than those used for BASE jumping
from VERY low altitude - about 600 ft or less) are equipped with
sliders to slow the opening to something only a little faster than
rounds. Square sport canopies often have sliders that slow the opening
enough to make it 'comfortable' but the emergency parachutes have just
enough slider to keep you from breaking your back in a terminal
velocity opening.

There are actually only two advantages to rounds.

The first is cost - old obsolete technology is always cheaper. You can
pick up a serviceable round rig for a couple hundred dollars US; a
square rig will be newer and more expensive.

The second is the reduced need for training. The square parachute is a
wing. It must be flown and flared for a landing. Further, for various
reasons I will be happy to go into if anyone is interested, it doesn't
really fly and flare like a rigid wing. Finding yourself under one
with 30 seconds to figure out its flight characteristics and land it in
a suboptimal landing area is NOT the hot tip. Much as I dislike rules,
I would still recommend a training jump or two for anyone planning to
use a square rig without prior experience, as well as briefing from
someone who understands both ram-air wing and rigid wing aerodynamics
and can prepare you for the differences.

Michael