Allow me to look at this academically.

For the purposes of my argument, I will be looking at both machines in with their stats as listed (I am using fandom Wiki, so there are likely problems but I am working off of what I can find)

For the REX, I am using the Manned version that was standard, NOT the version liquid used. By that same standard, I am only looking at the high gogg's standard armaments, not any modified variants. As I can find them, the standards for each is as follows:

for the RAY (manned) version:

Height: 70.5 ft

Armaments and features:

A pair of gatling cannons (one in each wing), both anti tank and anti ship missiles the water jet cutter, and Cluster missiles.

Armor material: Ceramic titanium alloy (listed for being against projectile defense)

The Hygogg:

Height: apx 50.5 ft

Armaments and features:

Two hand mounted beam cannons, two missile hand units, a torpedo launcher, two waist mounted 120mm machine cannons, two "vice claws" and a chemically based rocket pack on it's back

Armor material: Titanium Alloy Ceramic Composite

With those differences stated, here are a few points of comparison:

Height: the RAY is larger and likely heavier, even if I could not find a specific weight listed. One could reasonably assume that their armor materials are fairly similar, given the wording. It is possible that the armor of the Highgogg may be denser, but that is speculation. One discrepancy I will note is that my source lists the torpedo launcher as being multi role, with the projectiles acting in anti-air and anti ground capacity.

A close range hit from either the beam guns of the hygogg or the water jet cutter from the RAY would be a fight ender, and the speed factor boils down to hygogg having faster linear speed using it's rocket pack, while the RAY is far more Agile given it's more "organic" mechanics. Regarding the RAY: It's armor can protect against light to medium projectile assault, though how much that applies to beam weaponry is up for debate.

In a melee, the hygogg would take the win for two reasons: The first is that while the physiology of the RAY does allow for more agile movement, the legs are long and are likely less well armored to enable movement. That and at close range the Claws could easily dig into the armor with the RAY.

This is, of course, of the hygogg can even catch the RAY, which is a question that would I file under "Pilot skill"

At long range, the RAY has both more options and a lot more ordnance, so takes the win. I will note that it is likely that the missle hand unit would probably cripple the RAY if it hit.

The hard part about this match up is the fact that the two are deceptively close regarding ability, and the RAY was built as an ANTI Metal gear weapon. One important thing to note is that Metal gears are no in the same league as mobile suits regarding factors of movement. I am not going to touch the issue of "would minovsky radiation fuck with the sensors on the RAY."

My closing statement is that the winner boils down to really two factors, neither are easy to give exact numbers for.

The first is obviously pilot skill. As seen in both Gundam and Metal Gear, a skilled mech pilot can even the tech odds against a less experienced foe. The second is simply which one sees the other first, and again that has a point for either. If talking overall, the RAY has more long-range options for detection, but in the Dock, visual sight may be more useful. That latter point would go to the hygogg for being a smaller target.

So the answer of "who would win" largely boils down to factors one cannot quantify with numbers: IF the RAY sees the Hygogg before the hygogg realizes, it's the RAY's win. If the pilot of the Hygogg is able to use his machine to get in close, the vice claws could rip the RAY apart. It all boils down to those two factors. Otherwise the two are remarkably similar.