So, in my career of 5 years of playing guitar, a fairly significant upset has occurred that I've never dealt with before.
I've pretty much broken 2 strings not even within 24 hours of each other.

Both of them have broken at the bridge (1st it was my 3rd string, then my 4th), and my strings aren't even that worn either. I have two theories as to why they are breaking and I'm wondering what I can do to try and not have that happen again

1. I am palm muting too hard/ in the wrong place.

Recently I've been playing a lot of Damien Rice, and he has some pieces with some really percussive kinds of palm muting.

I palm mute by putting the fleshy part underneath my pink almost on top of the saddle, keeping the momentum of the strum, so there is a pretty significant amount of force, but it gets the sound I want

Since both strings broke at the bridge I could be putting lots of stress there creating a much higher likeliness for them to break.

2. I am simply strumming too hard

I use medium picks and for what I play there are some parts where I am consistently strumming really hard to keep the intensity of the song, at the apex of my strum, it's above the top edge of the guitar, and at the bottom, its pretty far below the pick guard, not to mention the frequent double strums I add in at the same volume.

I'm wondering if I could get higher gauge strings (I use Medium Elixir Phosphor Bronze's right now) or something like that, or if I have to do something with my technique
If the strings are breaking at the bridge, there is probably a small burr that is responsible.
There's no such thing; there never was. Where I am going you cannot follow me now.
Two strings really don't prove much, if 3 or 4 more break in no time, I'd worry about it.

Check for burrs, it might take a magnifying glass, or just fold some 400 grit sandpaper and give the saddle a quick touch up. Won't hurt, but I wouldn't be too worried unless it becomes a regular problem. I've seen brand new strings break in a day plenty times in 50 years playing.
Hmmm...I wonder what this button does...