I bought a Boss ME-70 pedal, and it all works 100% perfectly, but whenever I turn up the distortion I get this INSANE amount of buzzing/humming. I've cranked the noise suppressor and that does help quite a lot, but the buzz is still no where near quiet. I've tried switching cords around, and I even had my friend come over and we tried his guitar and amp with the pedal but it still did the same thing. I can't seem to figure this out..

(My guitar is a crappy Squier Bullet Strat and my amp is a custom tube my uncle made me. My friends guitar is a random no name Strat copy, and his amp is a Line 6 spyder 15/30... I don't remember exactly)

I've read posts online about how Humbuckers help reduce noise, and better quality guitars, and cords in general. Is my problem my low budget rig?
Your problem is the Squier Strat, because they have single coil pickups in them hum always comes with it on distortion. Having humbuckers reduces hum (and personally i think they're more versatile) because there are two coils to cancel out the hum. Notice on your squier if you flick it up to the 2nd position it uses the back and the middle and cancells the hum.
So basically yes, your problem is the guitar, the rest of your rig is quite good.
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Is my problem my low budget rig?

Your problem is not your low budget rig, I have a Fender Jazzmaster that I play on an orange rockerverb 50 combo amp with a wicked planet waves chord and i still get buzz. single coil soapbar pickups. although I DO notice that the buzz gets less intense as i get further away from the amp, so maybe try getting a longer patchcord and distancing yourself from the amp
Thanks for your answers I haven't tried humbuckers but that may be the problem. But it just doesn't seem right that my guitar should make THAT MUCH of a buzz because of humbuckers. I've seen youtube videos with people with my same guitar jamming with TONS of gain, right beside their amps.

I was reading online that nearby electronics could interfere with it as well. I have a TV, xbox, speakers, alarm clock, keyboard, and dimmer lights within about 15/20 square feet.

Now that I think about it, could part of the problem also be that I play with tons of gain while my amp isn't at a high volume? My parents aren't particularly fond of Heavy Metal, so I keep my amp's volume pretty much as low as possible while CRANKiNG the gain.

I'm just kinda throwing out ideas now. I don't have the money to buy a new guitar so I'm trying to make do with what I have haha.